To find the expression that is equal to [tex]\(\frac{4p^2 + 32p}{p-2} \cdot \frac{p^2 + p - 6}{2p^3 + 16p^2}\)[/tex], follow these detailed steps to simplify it:
1. Factorize each part of the expression:
- For the numerator [tex]\(4p^2 + 32p\)[/tex]:
[tex]\[
4p^2 + 32p = 4p(p + 8)
\][/tex]
- For the denominator [tex]\(p - 2\)[/tex]:
(It is already in its simplest form.)
- For the numerator [tex]\(p^2 + p - 6\)[/tex]:
[tex]\[
p^2 + p - 6 = (p+3)(p-2)
\][/tex]
- For the denominator [tex]\(2p^3 + 16p^2\)[/tex]:
[tex]\[
2p^3 + 16p^2 = 2p^2(p + 8)
\][/tex]
2. Rewrite the given expression with the factored terms:
[tex]\[
\frac{4p(p + 8)}{p - 2} \cdot \frac{(p + 3)(p - 2)}{2p^2(p + 8)}
\][/tex]
3. Cancel out common factors:
- The term [tex]\((p - 2)\)[/tex] appears in both the numerator and the denominator and thus can be cancelled out.
- The term [tex]\((p + 8)\)[/tex] also appears in both the numerator and the denominator and thus can be cancelled out.
[tex]\[
\frac{4p \cancel{(p + 8)}}{\cancel{p - 2}} \cdot \frac{(p + 3)\cancel{(p - 2)}}{2p^2 \cancel{(p + 8)}} = \frac{4p}{1} \cdot \frac{(p + 3)}{2p^2}
\][/tex]
4. Combine and simplify the remaining terms:
[tex]\[
\frac{4p(p + 3)}{2p^2} = \frac{4(p + 3)}{2p} = \frac{2(p + 3)}{p}
\][/tex]
5. Final simplified expression:
This shows that the given expression simplifies to:
[tex]\[
\boxed{\frac{2(p+3)}{p}}
\][/tex]
After following these steps, we find that the expression equal to [tex]\(\frac{4p^2 + 32p}{p-2} \cdot \frac{p^2 + p - 6}{2p^3 + 16p^2}\)[/tex] is [tex]\(\frac{2(p+3)}{p}\)[/tex].
