To find the product of the binomials [tex]\(\left(-2d^2 + s\right)\left(5d^2 - 6s\right)\)[/tex], we can use the distributive property (also known as the FOIL method for binomials). Let’s go through the steps:
1. First terms: Multiply the first terms in each binomial: [tex]\(-2d^2 \cdot 5d^2\)[/tex].
[tex]\[
-2d^2 \cdot 5d^2 = -10d^4
\][/tex]
2. Outer terms: Multiply the outer terms in each binomial: [tex]\(-2d^2 \cdot -6s\)[/tex].
[tex]\[
-2d^2 \cdot -6s = 12d^2s
\][/tex]
3. Inner terms: Multiply the inner terms in each binomial: [tex]\(s \cdot 5d^2\)[/tex].
[tex]\[
s \cdot 5d^2 = 5d^2s
\][/tex]
4. Last terms: Multiply the last terms in each binomial: [tex]\(s \cdot -6s\)[/tex].
[tex]\[
s \cdot -6s = -6s^2
\][/tex]
Now, add all these products together:
[tex]\[
-10d^4 + 12d^2s + 5d^2s - 6s^2
\][/tex]
Combine the like terms ([tex]\(12d^2s\)[/tex] and [tex]\(5d^2s\)[/tex]):
[tex]\[
-10d^4 + (12d^2s + 5d^2s) - 6s^2
\][/tex]
[tex]\[
-10d^4 + 17d^2s - 6s^2
\][/tex]
So, the product of [tex]\(\left(-2d^2 + s\right) \left(5d^2 - 6s\right)\)[/tex] is:
[tex]\[
\boxed{-10d^4 + 17d^2s - 6s^2}
\][/tex]
Therefore, the correct answer is:
[tex]\[
-10 d^4 + 17 d^2 s - 6 s^2
\][/tex]
