What is the product?

[tex]\[
\left(-2 d^2 + s \right) \left( 5 d^2 - 6 s \right)
\][/tex]

A. [tex]\(-10 d^4 + 17 d^2 s - 6 s^2\)[/tex]

B. [tex]\(-10 d^4 + 17 d^4 s^2 - 6 s^2\)[/tex]

C. [tex]\(-10 d^4 - 7 d^2 s - 6 s^2\)[/tex]

D. [tex]\(-10 d^4 + 17 d^2 s + 6 s^2\)[/tex]



Answer :

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]