To find the product of the expressions [tex]\((-2 a^2 + s)\)[/tex] and [tex]\((5 a^2 - 6 s)\)[/tex], we can perform polynomial multiplication. Let's break it down step-by-step.
Given:
[tex]\[
\left(-2 a^2 + s\right)\left(5 a^2 - 6 s\right)
\][/tex]
Step 1: Distribute each term in the first expression to each term in the second expression.
[tex]\[
= \left(-2 a^2 \cdot 5 a^2\right) + \left(-2 a^2 \cdot -6 s\right) + \left(s \cdot 5 a^2\right) + \left(s \cdot -6 s\right)
\][/tex]
Step 2: Simplify each of these products.
1. [tex]\(-2 a^2 \cdot 5 a^2 = -10 a^4\)[/tex]
2. [tex]\(-2 a^2 \cdot -6 s = 12 a^2 s\)[/tex]
3. [tex]\(s \cdot 5 a^2 = 5 a^2 s\)[/tex]
4. [tex]\(s \cdot -6 s = -6 s^2\)[/tex]
Step 3: Combine all the terms.
[tex]\[
= -10 a^4 + 12 a^2 s + 5 a^2 s - 6 s^2
\][/tex]
Step 4: Combine like terms.
[tex]\[
= -10 a^4 + (12 a^2 s + 5 a^2 s) - 6 s^2
= -10 a^4 + 17 a^2 s - 6 s^2
\][/tex]
Thus, the product of the expressions [tex]\((-2 a^2 + s)\)[/tex] and [tex]\((5 a^2 - 6 s)\)[/tex] is:
[tex]\[
-10 a^4 + 17 a^2 s - 6 s^2
\][/tex]
So, the correct answer is:
[tex]\[
-10 a^4 + 17 a^2 s - 6 s^2
\][/tex]
