Multiplying Polynomials Answer Key - Practice problems with solutions for algebra.

It seems you've uploaded an image containing a worksheet titled "Multiplying Polynomials ANSWER KEY." The task involves solving polynomial multiplication problems. Below, I will solve one of the problems step by step and explain the solution process.

Problem to Solve:

Let's take Problem 1 from the worksheet:
$$
5x(2x + 3) = ?
$$

Solution:

To solve this problem, we need to distribute the term \(5x\) across the terms inside the parentheses \((2x + 3)\). This is done using the distributive property of multiplication over addition, which states:
$$
a(b + c) = ab + ac
$$

Here, \(a = 5x\), \(b = 2x\), and \(c = 3\). Applying the distributive property:
$$
5x(2x + 3) = 5x \cdot 2x + 5x \cdot 3
$$

#### Step 1: Multiply \(5x \cdot 2x\)
- When multiplying terms with variables, multiply the coefficients and add the exponents of the same base.
- Coefficients: \(5 \cdot 2 = 10\)
- Variables: \(x \cdot x = x^2\)
So,
$$
5x \cdot 2x = 10x^2
$$

#### Step 2: Multiply \(5x \cdot 3\)
- Here, there is no variable in the second term, so it's simply:
$$
5x \cdot 3 = 15x
$$

#### Step 3: Combine the results
Now, combine the two results from Steps 1 and 2:
$$
5x(2x + 3) = 10x^2 + 15x
$$

Final Answer:

$$
\boxed{10x^2 + 15x}
$$

Explanation:

The distributive property is key here. Each term inside the parentheses is multiplied by the term outside the parentheses. After distributing, we simplify each product and combine like terms (if any). In this case, there were no like terms to combine, so the final answer is simply the sum of the two products.

If you have any specific questions or need further clarification, feel free to ask!