Simplify (mn)(x) = (x^2 + 4x)(x): Step-by-Step Solution

Hey guys! Let's dive into this math problem together. We've got the expression $(m \cdot n)(x) = (x^2 + 4x)(x)$, and our mission is to figure out which of the given options it's equal to. Don't worry, we'll break it down step by step so it's super clear. So, grab your pencils, and let's get started!

Understanding the Expression (mn)(x)

Okay, so the expression we're working with is $(m \cdot n)(x) = (x^2 + 4x)(x)$. What this means is that we're going to take the expression $(x^2 + 4x)$ and multiply it by $x$. This is a classic example of the distributive property in action. We need to make sure that each term inside the parentheses gets multiplied by the term outside. So, essentially, we need to distribute the $x$ across both $x^2$ and $4x$. When tackling problems like this, it’s super important to pay close attention to the order of operations and the rules of exponents. A small mistake in these areas can lead to a completely different answer. We'll take our time and make sure we get it right. Remember, in mathematics, precision is key, and understanding the underlying principles will always lead you to the correct solution. Think of it like building a house – a solid foundation ensures everything else stands strong. In this case, our foundation is the distributive property and the laws of exponents. Let's get to work and see how this unfolds!

Applying the Distributive Property

The key to simplifying this expression lies in the distributive property. Remember, this property states that $a(b + c) = ab + ac$. In our case, $a$ is $x$, $b$ is $x^2$, and $c$ is $4x$. So, we need to multiply $x$ by both $x^2$ and $4x$.

Let's break it down:

• $x * x^2 = x^3$ (Remember, when you multiply variables with exponents, you add the exponents. So, $x^1 * x^2 = x^{1+2} = x^3$)
• $x * 4x = 4x^2$ (Here, we're multiplying $x$ by $4x$. So, $1 * 4 = 4$ and $x^1 * x^1 = x^{1+1} = x^2$)

So, when we apply the distributive property, we get:

$(x^2 + 4x)(x) = x * x^2 + x * 4x = x^3 + 4x^2$

This step is crucial because it transforms the original expression into a more simplified form. It’s like taking a complicated puzzle and starting to piece it together. Each multiplication we perform brings us closer to the final solution. Always double-check your work at this stage to ensure you haven't made any small errors, as these can propagate through the rest of the problem. And remember, practice makes perfect! The more you work with the distributive property, the more comfortable and confident you'll become. Keep going, guys – we’re making great progress!

Checking the Options

Now that we've simplified the expression, let's compare our result, $x^3 + 4x^2$, to the given options:

A. $5x^2$ B. $x^3 + 4x^2$ C. $4x^4$

It's clear that our simplified expression, $x^3 + 4x^2$, matches option B. Therefore, the correct answer is B. This part of the problem is like the final piece of the puzzle slotting into place. We've done the hard work of simplifying the expression, and now we can confidently select the correct answer. It's always a good feeling when everything comes together! But don't rush through this step. Take a moment to ensure that you're comparing the correct terms and that you haven't missed anything. Double-checking is a sign of a careful mathematician, and it’s a great habit to cultivate. So, with a clear head and confident steps, let's move forward!

Detailed Explanation of Each Incorrect Option

To fully understand why option B is the correct answer, it's beneficial to examine why the other options are incorrect. This not only reinforces the correct method but also helps in avoiding similar mistakes in the future. Let's break down each incorrect option:

Option A: $5x^2$

This option, $5x^2$, is incorrect because it seems like an attempt to add the terms $x^2$ and $4x$ directly, but without correctly applying the distributive property and multiplying by $x$. It’s a common mistake to try and combine terms that aren’t like terms, but in this case, we’re multiplying, not adding, inside the parentheses initially. If we were to arrive at $5x^2$, it would imply a misunderstanding of the order of operations and the distributive property. Remember, we first need to multiply $x$ by each term inside the parentheses before we can simplify further. This highlights the critical importance of following the correct mathematical procedures. Math isn't just about getting to the answer; it's about the journey and the logical steps we take along the way. So, understanding why $5x^2$ is wrong helps us appreciate the correct process even more.

Option C: $4x^4$

Option C, $4x^4$, is incorrect because it likely results from multiplying the exponents incorrectly. If someone mistakenly multiplied $x^2$ by $4x$ and then squared the result, they might arrive at this answer. However, we need to remember that we are distributing $x$ across the terms $x^2$ and $4x$, not multiplying the terms inside the parentheses by each other initially. This mistake emphasizes the importance of the correct application of exponent rules and the distributive property. It's essential to understand how exponents work and when to add them versus multiply them. Mistakes like this are opportunities to learn and refine our understanding of the fundamental rules of algebra. So, by identifying why $4x^4$ is incorrect, we’re reinforcing our knowledge of these key concepts and building a stronger foundation for future problems.

Key Takeaways and Tips

Let's recap the main points and offer some helpful tips for tackling similar problems in the future:

1. Master the Distributive Property: This is a foundational concept in algebra. Make sure you understand how to apply it correctly. Remember, $a(b + c) = ab + ac$.
2. Understand Exponent Rules: When multiplying variables with exponents, you add the exponents (e.g., $x^m * x^n = x^{m+n}$). A clear understanding of these rules is crucial.
3. Follow the Order of Operations: Always remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). This helps you approach problems in the correct sequence.
4. Double-Check Your Work: It's always a good idea to review your steps and calculations to minimize errors. A small mistake early on can lead to an incorrect final answer.
5. Practice Regularly: The more you practice, the more comfortable you'll become with these concepts. Try solving similar problems to reinforce your understanding.

By keeping these takeaways in mind, you'll be well-equipped to handle algebraic expressions and equations with confidence. Math is like a muscle – the more you exercise it, the stronger it gets. So, keep practicing and keep learning!

Conclusion

So, to wrap things up, when we simplify the expression $(m \cdot n)(x) = (x^2 + 4x)(x)$, the correct answer is B. $x^3 + 4x^2$. We arrived at this answer by carefully applying the distributive property and the rules of exponents. By understanding the underlying principles and avoiding common mistakes, we can confidently solve similar problems. Remember, math isn't just about finding the right answer; it's about understanding the process and the logic behind it. Keep practicing, keep asking questions, and most importantly, keep enjoying the journey of learning! You guys got this! Keep up the great work, and I'll see you in the next math adventure!