Sure, let's simplify the given expression step-by-step.
We start with the algebraic fraction:
[tex]\[
\frac{3 p^3 + 15 p^4 - 9 p^3}{3 p^2}
\][/tex]
First, let's combine like terms in the numerator:
[tex]\[
3 p^3 - 9 p^3 + 15 p^4 = (3 p^3 - 9 p^3) + 15 p^4 = -6 p^3 + 15 p^4
\][/tex]
So now the expression is:
[tex]\[
\frac{-6 p^3 + 15 p^4}{3 p^2}
\][/tex]
Next, let's factor out common terms in the numerator:
[tex]\[
-6 p^3 + 15 p^4 = 3 p^3 (-2 + 5 p)
\][/tex]
Now we substitute this back into the expression:
[tex]\[
\frac{3 p^3 (-2 + 5 p)}{3 p^2}
\][/tex]
We can cancel out the common factor of [tex]\(3 p^2\)[/tex] from both the numerator and the denominator. Since [tex]\(3 p^3\)[/tex] in the numerator is [tex]\(3 p^2 \cdot p\)[/tex], we get:
[tex]\[
\frac{3 p^3 (-2 + 5 p)}{3 p^2} = \frac{3 p^2 \cdot p (-2 + 5 p)}{3 p^2} = p (-2 + 5 p)
\][/tex]
Simplify the expression one last time:
[tex]\[
p (-2 + 5 p) = -2p + 5p^2
\][/tex]
So, the simplified expression is:
[tex]\[
\boxed{-2p + 5p^2}
\][/tex]
