To multiply the given rational expressions:
[tex]\[
\frac{p^9 q}{p^2 + p - 12} \cdot \frac{2p^2 - 11p + 15}{p q^9}
\][/tex]
we follow these steps:
1. Factor the denominators and numerators where possible.
- For the denominator \( p^2 + p - 12 \):
[tex]\[
p^2 + p - 12 = (p + 4)(p - 3)
\][/tex]
- For the numerator \( 2p^2 - 11p + 15 \):
[tex]\[
2p^2 - 11p + 15 = (2p - 5)(p - 3)
\][/tex]
Now the expressions look as follows:
[tex]\[
\frac{p^9 q}{(p + 4)(p - 3)} \cdot \frac{(2p - 5)(p - 3)}{p q^9}
\][/tex]
2. Combine the two fractions into one:
[tex]\[
\frac{p^9 q \cdot (2p - 5)(p - 3)}{(p + 4)(p - 3) \cdot p q^9}
\][/tex]
3. Cancel out common factors in the numerator and the denominator:
- The factor \(p - 3\) appears in both the numerator and the denominator and can be canceled out.
- The \(p\) term in the numerator and denominator can be partially canceled. Specifically, \(p^9\) in the numerator and \(p\) in the denominator leave \(p^{9-1} = p^8\).
- The \(q\) term in the numerator and denominator can be partially canceled. Specifically, \(q\) in the numerator and \(q^9\) in the denominator leave \(q^{9-1} = q^8\).
Simplifying:
[tex]\[
\frac{p^8 \cdot (2p - 5)}{(p + 4) \cdot q^8}
\][/tex]
4. Write the simplified form.
The result is:
[tex]\[
\frac{p^8 (2p - 5)}{(p + 4) q^8}
\][/tex]
Therefore, the simplified form of the product of the given rational expressions is:
[tex]\[
\boxed{\frac{p^8 (2p - 5)}{(p + 4) q^8}}
\][/tex]
