Example of multiplying polynomials from KutaSoftware Algebra 1, illustrating 6v(2v + 3) = 12v² + 18v using the distributive property.

Let's solve the problem shown in the image step by step and explain the solution.

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Problem:

Multiply the polynomial:
$$
6v(2v + 3)
$$

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Step 1: Understand the Expression

We are multiplying a monomial $6v$ by a binomial $(2v + 3)$. This requires distributing the $6v$ to each term inside the parentheses.

This uses the Distributive Property:
$$
a(b + c) = ab + ac
$$

So:
$$
6v(2v + 3) = 6v \cdot 2v + 6v \cdot 3
$$

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Step 2: Multiply Each Term

#### First term: $6v \cdot 2v$

- Multiply the coefficients: $6 \cdot 2 = 12$
- Multiply the variables: $v \cdot v = v^2$ (since $v^1 \cdot v^1 = v^{1+1} = v^2$)

So:
$$
6v \cdot 2v = 12v^2
$$

#### Second term: $6v \cdot 3$

- Multiply the coefficient: $6 \cdot 3 = 18$
- The variable remains $v$

So:
$$
6v \cdot 3 = 18v
$$

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Step 3: Combine the Results

Now add the two terms:
$$
12v^2 + 18v
$$

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✔ Final Answer:

$$
\boxed{12v^2 + 18v}
$$

This matches the answer shown in the image, which is correctly boxed in red.

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🔍 Additional Notes from the Image:

- The yellow arrows show that $6v$ is being distributed to both $2v$ and $3$.
- On the right, there’s a reminder of exponent rules:
- $a^x \cdot a^y = a^{x+y}$
- So $v \cdot v = v^{1+1} = v^2$

These rules help justify why $6v \cdot 2v = 12v^2$.

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✔ Conclusion: The multiplication is done correctly using distribution and exponent rules.
Answer: $12v^2 + 18v$