We are asked to factorize the expression $(x^2 + 4x)^2 + 6(x^2 + 4x) + 8$.

1. Problem Description

We are asked to factorize the expression $(x^2 + 4x)^2 + 6(x^2 + 4x) + 8$.

2. Solution Steps

Let $y = x^2 + 4x$. Then the expression becomes $y^2 + 6y + 8$.

We can factorize this quadratic expression as follows:

$y^2 + 6y + 8 = (y+2)(y+4)$

Now, substitute $y = x^2 + 4x$ back into the expression:

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

Notice that $x^2 + 4x + 4$ can be further factored as $(x+2)^2$.

Thus we have

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

3. Final Answer

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