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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 this problem step-by-step, we need to transform the given mathematical statement while providing justifications for each step.

1. Original Mathematical Statement:
[tex]\[ -2(2x^3 + 4x^2 - 3) + 5(x^2 - 2x - 2) \][/tex]

2. Step 1: Apply the Distributive Property.
The distributive property states that [tex]\( a(b + c) = ab + ac \)[/tex]. We will distribute [tex]\(-2\)[/tex] and [tex]\(5\)[/tex] across the terms in the parentheses:

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

Simplifying each term gives us:
[tex]\[ -4x^3 - 8x^2 + 6 + 5x^2 - 10x - 10 \][/tex]

Justification: Distributive Property

3. Step 2: Apply the Commutative Property of Addition.
The commutative property of addition states that [tex]\( a + b = b + a \)[/tex]. We can reorder terms to group like terms together:

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

Justification: Commutative Property of Addition

4. Step 3: Combine Like Terms.
Now, we combine the like terms (terms with the same power of [tex]\( x \)[/tex]):

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

Simplifying each group of like terms gives us:
[tex]\[ -4x^3 - 3x^2 - 10x - 4 \][/tex]

Justification: Combine Like Terms

Thus, the missing justifications in the table are:

- Distributive Property
- Commutative Property of Addition
- Combine Like Terms

Therefore, the correct order of justifications is:

Distributive Property; Commutative Property of Addition; Combine Like Terms