To determine the product of [tex]\((4s + 2)(5s^2 + 10s + 3)\)[/tex], we need to use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial. Let's break it down step-by-step:
1. Expand [tex]\((4s + 2)(5s^2 + 10s + 3)\)[/tex]:
- First, distribute [tex]\(4s\)[/tex] to each term in the second polynomial:
[tex]\[
4s \times 5s^2 = 20s^3
\][/tex]
[tex]\[
4s \times 10s = 40s^2
\][/tex]
[tex]\[
4s \times 3 = 12s
\][/tex]
- Now, distribute [tex]\(2\)[/tex] to each term in the second polynomial:
[tex]\[
2 \times 5s^2 = 10s^2
\][/tex]
[tex]\[
2 \times 10s = 20s
\][/tex]
[tex]\[
2 \times 3 = 6
\][/tex]
2. Combine all the resulting products:
[tex]\[
(20s^3 + 40s^2 + 12s) + (10s^2 + 20s + 6)
\][/tex]
3. Combine like terms:
[tex]\[
20s^3 + (40s^2 + 10s^2) + (12s + 20s) + 6
\][/tex]
[tex]\[
20s^3 + 50s^2 + 32s + 6
\][/tex]
Therefore, the product of [tex]\((4s + 2)(5s^2 + 10s + 3)\)[/tex] is:
[tex]\[
20s^3 + 50s^2 + 32s + 6
\][/tex]
So, the correct option is:
[tex]\[
20s^3 + 50s^2 + 32s + 6
\][/tex]
