Certainly! Here is a step-by-step solution to expand and simplify the expression [tex]\((a p^2 + 3 p q)(5 p + 3 q)\)[/tex].
Given the expression:
[tex]\[
(a p^2 + 3 p q)(5 p + 3 q)
\][/tex]
We will use the distributive property (also known as the FOIL method for binomials) to expand this product. This involves multiplying each term in the first expression by each term in the second expression:
1. First, distribute [tex]\(a p^2\)[/tex]:
[tex]\[
a p^2 \cdot 5 p = 5 a p^3
\][/tex]
[tex]\[
a p^2 \cdot 3 q = 3 a p^2 q
\][/tex]
2. Next, distribute [tex]\(3 p q\)[/tex]:
[tex]\[
3 p q \cdot 5 p = 15 p^2 q
\][/tex]
[tex]\[
3 p q \cdot 3 q = 9 p q^2
\][/tex]
Now, combine all these results:
[tex]\[
(a p^2 + 3 p q)(5 p + 3 q) = 5 a p^3 + 3 a p^2 q + 15 p^2 q + 9 p q^2
\][/tex]
Thus, the expanded form of the given expression is:
[tex]\[
5 a p^3 + 3 a p^2 q + 15 p^2 q + 9 p q^2
\][/tex]
This is the simplified and expanded result.
