Question

When solving an equation, Carmen's first step is shown below. Which property justifies Carmen's first step?
Original Equation:
Original Equation:
10, left parenthesis, minus, 4, x, squared, minus, 4, right parenthesis, plus, 5, equals, minus, 4, x, squared, minus, 10
10(−4x
2
−4)+5=
−4x
2
−10
First Step:
First Step:
10, left parenthesis, minus, 4, x, squared, minus, 4, right parenthesis, equals, minus, 4, x, squared, minus, 15
10(−4x
2
−4)=
−4x
2
−15

Answer 1

Carmen's first step can be justified by using the **Distributive Property**.

In the original equation, Carmen is distributing the 10 to both terms inside the parentheses. Here's how it breaks down:

1. The expression \( 10(-4x^2 - 4) \) can be expanded using the Distributive Property:
\[
10(-4x^2) + 10(-4) = -40x^2 - 40
\]

2. When combined with the \( +5 \) from the original equation:
\[
-40 + 5 = -35
\]
Thus, we have:
\[
-40x^2 - 35
\]

So, the statement \( 10(-4x^2 - 4) = -40x^2 - 40 \) and the following adding of 5 to -40 gives you -15, supporting the conclusion that:
\[
10(-4x^2 - 4) + 5 = -4x^2 - 15
\]

Therefore, the correct property that justifies Carmen's first step is the **Distributive Property**.