To fill in the missing justifications in the correct order, let's consider the operations being performed in each step of the mathematical statement.
1. First, we need to distribute the coefficients -2 and 5 to the terms inside their respective parentheses. This uses the Distributive Property.
2. Next, we rearrange the terms to make it easier to combine like terms. This uses the Commutative Property of Addition.
3. Finally, we combine the like terms. This is the step where we simplify the expression by adding or subtracting the coefficients of the same powers of [tex]\(x\)[/tex].
So the correct order of justifications is as follows:
\begin{tabular}{|c|l|}
\hline 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}
Thus, the correct answer is: Distributive Property, Commutative Property of Addition; Combine Like Terms.
