Factor completely; simplify if possible. \( 4 x^{3}+32 x^{2}+64 x= \)

To factor the expression \( 4x^3 + 32x^2 + 64x \), start by factoring out the greatest common factor, which is \( 4x \): \[ 4x(x^2 + 8x + 16) \] Next, focus on the quadratic expression \( x^2 + 8x + 16 \). This can be factored further because it is a perfect square trinomial: \[ x^2 + 8x + 16 = (x + 4)^2 \] Thus, the fully factored form becomes: \[ 4x(x + 4)^2 \] Now, you have a completely factored expression: \( 4x(x + 4)^2 \).