The problem asks us to factor the polynomial $x^3 + 11x^2 + 30x$ completely, if possible. If it cannot be factored, we should state that the polynomial is prime.

1. Problem Description

The problem asks us to factor the polynomial $x^3 + 11x^2 + 30x$ completely, if possible. If it cannot be factored, we should state that the polynomial is prime.

2. Solution Steps

First, we look for a common factor in all the terms of the polynomial. We can see that $x$ is a common factor.

Factoring out $x$ gives us:

$x^3 + 11x^2 + 30x = x(x^2 + 11x + 30)$.

Now, we need to factor the quadratic expression $x^2 + 11x + 30$.

We are looking for two numbers that multiply to 30 and add up to

1

1. These numbers are 5 and 6, since $5 \cdot 6 = 30$ and $5 + 6 = 11$. Thus, we can factor the quadratic as:

$x^2 + 11x + 30 = (x + 5)(x + 6)$.

Putting it all together, the factored form of the original polynomial is:

$x^3 + 11x^2 + 30x = x(x + 5)(x + 6)$.

3. Final Answer

$x(x+5)(x+6)$