To factor the expression \( x^4 + 8x^3 + 15x^2 \), we first need to identify the greatest common factor (GCF) of the three terms.
The given expression can be rewritten as:
\[ x^2(x^2 + 8x + 15) \]
Here, the GCF is \( x^2 \). Now we will complete the factorization by focusing on the quadratic expression \( x^2 + 8x + 15 \).
Next, we factor the quadratic. We need to find two numbers that multiply to \( 15 \) (the constant term) and add to \( 8 \) (the coefficient of the linear term). The numbers \( 3 \) and \( 5 \) satisfy these conditions since:
\[ 3 \times 5 = 15 \quad \text{and} \quad 3 + 5 = 8 \]
Thus, we can factor the quadratic as follows:
\[ x^2 + 8x + 15 = (x + 3)(x + 5) \]
Putting it all together, the complete factorization of the original expression is:
\[ x^4 + 8x^3 + 15x^2 = x^2(x + 3)(x + 5) \]
So, the final answer is:
\[ \boxed{x^2(x + 3)(x + 5)} \]