The problem asks to factor the polynomial $x^3 + 3x^2 + 5x + 15$ completely.

1. Problem Description

The problem asks to factor the polynomial $x^3 + 3x^2 + 5x + 15$ completely.

2. Solution Steps

We can try to factor the polynomial by grouping.

$x^3 + 3x^2 + 5x + 15 = (x^3 + 3x^2) + (5x + 15)$

From the first group $(x^3 + 3x^2)$, we can factor out $x^2$, which gives us $x^2(x + 3)$.

From the second group $(5x + 15)$, we can factor out $5$, which gives us $5(x + 3)$.

So, the expression becomes $x^2(x+3) + 5(x+3)$.

Now we can factor out $(x+3)$ from the entire expression:

$x^2(x+3) + 5(x+3) = (x+3)(x^2+5)$.

The quadratic $x^2 + 5$ has no real roots, because $x^2 + 5 = 0$ means $x^2 = -5$, so $x = \pm \sqrt{-5} = \pm i\sqrt{5}$, which are imaginary roots.

Thus, $x^2 + 5$ cannot be factored further using real numbers.

3. Final Answer

$(x+3)(x^2+5)$