Solving Polynomial Expressions: A Step-by-Step Guide

Hey everyone! Let's dive into a common math problem: finding the equivalent expression for (x + 3)(-2x² + 2x + 5). This is a classic example of multiplying polynomials, and it's super important for algebra. We're going to break it down step-by-step so you can totally nail it. We will use the original input, which is a great starting point for understanding how to approach and solve this type of problem. So, grab your pencils and let's get started!

Understanding the Problem: Expanding Polynomials

Okay, so the main goal here is to expand the expression. What does that mean, exactly? Well, expanding means multiplying out all the terms. We've got a binomial, (x + 3), multiplied by a trinomial, (-2x² + 2x + 5). To solve this, we need to use the distributive property. Remember, the distributive property tells us that we multiply each term in the first set of parentheses by each term in the second set of parentheses. This is a crucial concept in algebra, so understanding it is going to set you up for success in more complex equations. Polynomials are the foundation in algebra. Mastering this will make all the difference, so focus up and let's get this done.

Let's get into the details, shall we? You'll be using the distributive property, which is just a fancy way of saying we're going to multiply each term in the first set of parentheses by each term in the second. Think of it like this: the x in (x + 3) needs to be multiplied by each term in (-2x² + 2x + 5), and then the 3 in (x + 3) needs to be multiplied by each term in (-2x² + 2x + 5). It might seem like a lot, but trust me, it’s straightforward once you get the hang of it. Keeping everything organized is the key to getting the correct answer. The more organized you are, the less likely you are to make mistakes. So, write everything out clearly and take your time. You got this, guys!

So, let’s begin breaking this problem down. This type of equation, which deals with expanding polynomial expressions, is common in algebra. This kind of practice will help you build a solid understanding of algebraic concepts. Make sure to practice, practice, and practice some more. The more you do, the easier it gets, trust me! Take your time, break down the problem into smaller parts, and you'll find that expanding polynomials becomes a breeze. So, are you ready to get this thing done? Let's dive in and learn!

Step-by-Step Solution: Multiplying It Out

Alright, let’s get down to business and start solving this problem. First, we will be multiplying x by each term in the trinomial: x * (-2x²), x * (2x), and x * (5). This gives us -2x³ + 2x² + 5x. Then, we will take the 3 and multiply it by each term in the trinomial: 3 * (-2x²), 3 * (2x), and 3 * (5). This results in -6x² + 6x + 15. Now, we have two expressions, and all that's left is to combine these results. We’ll combine them by adding the two sets of terms together: (-2x³ + 2x² + 5x) + (-6x² + 6x + 15). This is where we will combine all the like terms. This is the stage where we combine terms with the same power of x. Remember that? Great!

Let's keep things moving, okay? Let’s combine like terms. This means we're looking for terms that have the same variable raised to the same power. This is where it all comes together! For the x³ term, we only have -2x³, so that stays as is. For the x² terms, we have 2x² and -6x². Combine those, and you get -4x². For the x terms, we have 5x and 6x. Adding those up gives us 11x. Finally, we have the constant term, 15, which doesn't have any other terms to combine with. So, we're left with -2x³ - 4x² + 11x + 15. This is what we were looking for! We have our simplified, expanded expression!

It might seem a bit daunting at first, but with practice, you’ll become super comfortable with it. Remember, practice is key! Keep doing problems like these, and you'll become a pro in no time. You will get more and more comfortable as you go, and you’ll find that you actually enjoy it. So let's keep going and finish this thing. It's time to choose the answer!

Identifying the Correct Answer: Matching the Expression

Now that we have expanded and simplified the expression (x + 3)(-2x² + 2x + 5) to -2x³ - 4x² + 11x + 15, it's time to check which of the multiple-choice options matches our answer. Going back to the question, the options are as follows:

A. -2x³ - 4x² + 11x + 15 B. -2x² + 3x + 8 C. -2x³ - 8x² + x + 15 D. -2x² + 2x + 15

By comparing our simplified expression, -2x³ - 4x² + 11x + 15, with the given options, we can see that Option A is the correct match. That’s because the expressions in option A are identical! Congratulations! You have successfully solved the problem and found the correct answer! Give yourself a pat on the back.

Now that we have reviewed all of the answers, this becomes a lot easier. In this case, we were able to quickly determine which one was correct. Just remember, the more practice you get, the easier this becomes. Now let's wrap this up!

Conclusion: Mastering Polynomial Multiplication

Alright, folks, we've reached the end of our journey through this polynomial multiplication problem! We started with (x + 3)(-2x² + 2x + 5), and through the power of the distributive property and combining like terms, we successfully expanded and simplified it to -2x³ - 4x² + 11x + 15. We then identified that Option A was the right answer. Now, you should feel confident in your ability to solve similar problems. If you stick to the steps, you'll be fine.

Key takeaways: Remember the distributive property, be careful when multiplying, and always combine like terms! Keep practicing these types of problems to become a pro! Keep up the good work. Keep learning, and keep growing. Do some more problems to challenge yourselves. You’ve got this. Never be afraid to ask for help or look back at the process. We will see you in the next one, and until then, keep learning!

This method is super useful for other polynomial problems as well. So, next time you see a problem like this, you will know exactly what to do. If you have any questions, feel free to ask in the comments. Keep up the great work, everyone! And with that, let's wrap this up, shall we?