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\begin{tabular}{|l|l|}
\hline \multicolumn{1}{|c|}{ Mathematical Statement } & Justification \\
\hline [tex]$-2\left(2 x^3+4 x^2-3\right)+5\left(x^2-2 x-2\right)$[/tex] & Given \\
\hline [tex]$-4 x^3-8 x^2+6+5 x^2-10 x-10$[/tex] & Distributive Property \\
\hline [tex]$-4 x^3-8 x^2+5 x^2-10 x+6-10$[/tex] & Commutative Property of Addition \\
\hline [tex]$-4 x^3-3 x^2-10 x-4$[/tex] & Combine Like Terms \\
\hline
\end{tabular}

Fill in the missing justifications in the correct order.

A. Combine Like Terms; Distributive Property; Commutative Property of Addition
B. Commutative Property of Addition; Combine Like Terms; Distributive Property
C. Distributive Property; Commutative Property of Addition; Combine Like Terms
D. Distributive Property; Combine Like Terms; Commutative Property of Addition



Answer :

GinnyAnswer
To solve the problem step by step:

1. First, we start with the given expression:
[tex]\[ -2(2x^3 + 4x^2 - 3) + 5(x^2 - 2x - 2) \][/tex]
This is simply the initial expression we are working with.

2. Next, we will apply the Distributive Property:

[tex]\[ -2 \cdot 2x^3 + (-2) \cdot 4x^2 + (-2)(-3) + 5 \cdot x^2 + 5 \cdot (-2x) + 5 \cdot (-2) \][/tex]

Which simplifies to:

[tex]\[ -4x^3 - 8x^2 + 6 + 5x^2 - 10x - 10 \][/tex]

3. Then, we Combine Like Terms to simplify the expression:

Combine [tex]\(-8x^2\)[/tex] and [tex]\(5x^2\)[/tex] to get [tex]\(-3x^2\)[/tex].

Combine [tex]\(6\)[/tex] and [tex]\(-10\)[/tex] to get [tex]\(-4\)[/tex].

So, we have:

[tex]\[ -4x^3 - 8x^2 + 5x^2 - 10x + 6 - 10 \][/tex]

Which simplifies to:

[tex]\[ -4x^3 - 3x^2 - 10x - 4 \][/tex]

4. Finally, we can use the Commutative Property of Addition to verify the order does not change the expression:

Rearranging terms would still give us:

[tex]\[ -4x^3 - 3x^2 - 10x - 4 \][/tex]

Hence, filling in the missing justifications, the correct order is:

Distributive Property; Combine Like Terms; Commutative Property of Addition

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