Sure! Let's work through the multiplication of the expressions step by step:
We start with the expressions:
[tex]\[ \left(-2 a x^2 + 3 a x - a^2\right) \][/tex]
and
[tex]\[ \left(-a^2 x^2\right). \][/tex]
To find the product of these two expressions, we need to distribute [tex]\(-a^2 x^2\)[/tex] across each term inside the parentheses of the first expression.
### Step-by-Step Distribution:
1. Multiply [tex]\(-a^2 x^2\)[/tex] by [tex]\(-2 a x^2\)[/tex]:
[tex]\[
\left(-a^2 x^2\right) \cdot \left(-2 a x^2\right) = -a^2 x^2 \cdot -2 a x^2
\][/tex]
[tex]\[
= (-a^2) \cdot (-2 a) \cdot x^2 \cdot x^2
\][/tex]
[tex]\[
= 2 a^3 x^4.
\][/tex]
2. Multiply [tex]\(-a^2 x^2\)[/tex] by [tex]\(3 a x\)[/tex]:
[tex]\[
\left(-a^2 x^2\right) \cdot \left(3 a x\right) = -a^2 x^2 \cdot 3 a x
\][/tex]
[tex]\[
= (-a^2) \cdot (3 a) \cdot x^2 \cdot x
\][/tex]
[tex]\[
= -3 a^3 x^3.
\][/tex]
3. Multiply [tex]\(-a^2 x^2\)[/tex] by [tex]\(-a^2\)[/tex]:
[tex]\[
\left(-a^2 x^2\right) \cdot \left(-a^2\right) = -a^2 x^2 \cdot -a^2
\][/tex]
[tex]\[
= (-a^2) \cdot (-a^2) \cdot x^2
\][/tex]
[tex]\[
= a^4 x^2.
\][/tex]
After performing all the multiplications, we add up all the resulting terms:
[tex]\[
2 a^3 x^4 - 3 a^3 x^3 + a^4 x^2.
\][/tex]
So, the product of the expressions [tex]\(\left(-2 a x^2 + 3 a x - a^2\right)\)[/tex] and [tex]\(\left(-a^2 x^2\right)\)[/tex] is:
[tex]\[
a^4 x^2 + 2 a^3 x^4 - 3 a^3 x^3.
\][/tex]
