To find the product of the given expressions [tex]\(\left(-2 d^2 + s\right)\left(5 d^2 - 6 s\right)\)[/tex], we will follow the distributive property (also known as the FOIL method for binomials). Here is the step-by-step solution:
1. Distribute Each Term in the First Expression to Each Term in the Second Expression:
[tex]\[
(-2 d^2 + s)(5 d^2 - 6 s)
\][/tex]
This means we will multiply each term in the first expression by each term in the second expression:
[tex]\[
(-2 d^2) \cdot (5 d^2) + (-2 d^2) \cdot (-6 s) + s \cdot (5 d^2) + s \cdot (-6 s)
\][/tex]
2. Multiply the Terms:
- First term: [tex]\((-2 d^2) \cdot (5 d^2)\)[/tex]
[tex]\[
-2 \cdot 5 \cdot d^4 = -10 d^4
\][/tex]
- Second term: [tex]\((-2 d^2) \cdot (-6 s)\)[/tex]
[tex]\[
-2 \cdot -6 \cdot d^2 \cdot s = 12 d^2 s
\][/tex]
- Third term: [tex]\(s \cdot (5 d^2)\)[/tex]
[tex]\[
5 s \cdot d^2 = 5 d^2 s
\][/tex]
- Fourth term: [tex]\(s \cdot (-6 s)\)[/tex]
[tex]\[
-6 s^2
\][/tex]
3. Combine Like Terms:
Add together all the resulting terms:
[tex]\[
-10 d^4 + 12 d^2 s + 5 d^2 s - 6 s^2
\][/tex]
Combine the [tex]\(d^2 s\)[/tex] terms:
[tex]\[
12 d^2 s + 5 d^2 s = 17 d^2 s
\][/tex]
4. Write the Final Simplified Expression:
[tex]\[
-10 d^4 + 17 d^2 s - 6 s^2
\][/tex]
Therefore, the correct product of the given expressions is:
[tex]\[
-10 d^4 + 17 d^2 s - 6 s^2
\][/tex]
