The problem asks to factor the expression $8x^4 + 52x^3 + 60x^2$ completely. First, find the greatest common factor (GCF) of the terms, factor it out, and then factor the remaining quadratic expression, if possible.

1. Problem Description

The problem asks to factor the expression $8x^4 + 52x^3 + 60x^2$ completely. First, find the greatest common factor (GCF) of the terms, factor it out, and then factor the remaining quadratic expression, if possible.

2. Solution Steps

First, find the GCF of the coefficients 8, 52, and

6

0. The prime factorization of 8 is $2^3$.

The prime factorization of 52 is $2^2 \cdot 13$.

The prime factorization of 60 is $2^2 \cdot 3 \cdot 5$.

The GCF of the coefficients is $2^2 = 4$.

Next, find the GCF of the variable terms $x^4$, $x^3$, and $x^2$. The smallest power of $x$ is $x^2$, so the GCF of the variable terms is $x^2$.

Thus, the greatest common factor of the entire expression is $4x^2$.

Factor out $4x^2$ from the expression:

$8x^4 + 52x^3 + 60x^2 = 4x^2(2x^2 + 13x + 15)$

Now, factor the quadratic expression $2x^2 + 13x + 15$. We are looking for two numbers that multiply to $2 \cdot 15 = 30$ and add to

1

3. The numbers 3 and 10 satisfy these conditions.

Rewrite the middle term:

$2x^2 + 13x + 15 = 2x^2 + 3x + 10x + 15$

Factor by grouping:

$2x^2 + 3x + 10x + 15 = x(2x + 3) + 5(2x + 3)$

Factor out the common binomial factor $(2x + 3)$:

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

Therefore, $2x^2 + 13x + 15 = (2x + 3)(x + 5)$.

Substitute this back into the expression with the GCF:

$4x^2(2x^2 + 13x + 15) = 4x^2(2x + 3)(x + 5)$

3. Final Answer

$4x^2(x+5)(2x+3)$