Certainly! Let's find the product of the two polynomials: [tex]\((4s + 2)(5s^2 + 10s + 3)\)[/tex].
To do this, we will use the distributive property (also known as the FOIL method in the case of binomials) to multiply each term in the first polynomial by each term in the second polynomial.
1. Start by distributing [tex]\(4s\)[/tex] across the second polynomial:
[tex]\[
4s \cdot (5s^2 + 10s + 3) = (4s \cdot 5s^2) + (4s \cdot 10s) + (4s \cdot 3)
\][/tex]
[tex]\[
= 20s^3 + 40s^2 + 12s
\][/tex]
2. Next, distribute [tex]\(2\)[/tex] across the second polynomial:
[tex]\[
2 \cdot (5s^2 + 10s + 3) = (2 \cdot 5s^2) + (2 \cdot 10s) + (2 \cdot 3)
\][/tex]
[tex]\[
= 10s^2 + 20s + 6
\][/tex]
3. Now, add the results from steps 1 and 2 together:
[tex]\[
20s^3 + 40s^2 + 12s + 10s^2 + 20s + 6
\][/tex]
4. 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].
The correct answer is:
[tex]\[
20s^3 + 50s^2 + 32s + 6
\][/tex]
