Let's examine the given mathematical statement step-by-step and fill in the justifications.
Given Statement:
[tex]\[
-2\left(2 x^3+4 x^2-3\right)+5\left(x^2-2 x-2\right)
\][/tex]
Step 1:
First, we apply the distributive property, which allows us to distribute the constants [tex]\(-2\)[/tex] and [tex]\(5\)[/tex] to each term inside the parentheses:
[tex]\[
-2(2 x^3) + (-2)(4 x^2) + (-2)(-3) + 5(x^2) + 5(-2 x) + 5(-2)
\][/tex]
This simplifies to:
[tex]\[
-4 x^3 - 8 x^2 + 6 + 5 x^2 - 10 x - 10
\][/tex]
Justification for Step 1:
Distributive Property
Step 2:
Next, we rearrange the terms while preserving their signs to group like terms together. This step does not change the mathematical expression but helps to make combining terms easier:
[tex]\[
-4 x^3 - 8 x^2 + 5 x^2 - 10 x + 6 - 10
\][/tex]
Justification for Step 2:
Commutative Property of Addition
Step 3:
Finally, we combine like terms by adding or subtracting coefficients of the same power of [tex]\(x\)[/tex]:
[tex]\[
-4 x^3 + (-8 x^2 + 5 x^2) + (-10 x) + (6 - 10)
\][/tex]
This simplifies to:
[tex]\[
-4 x^3 - 3 x^2 - 10 x - 4
\][/tex]
Justification for Step 3:
Combine Like Terms
Thus, the missing justifications are, in order:
[tex]\[
\boxed{\text{Distributive Property, Commutative Property of Addition, Combine Like Terms}}
\][/tex]
