To factor the expression \(4x^4 - 24x^3 + 36x^2\) completely, we can start by identifying the greatest common factor (GCF).
- The GCF of the coefficients (4, -24, and 36) is 4.
- The lowest power of \(x\) in all terms is \(x^2\).
So, we can factor out \(4x^2\):
\[ 4x^2(x^2 - 6x + 9) \]
Next, we will factor the quadratic \(x^2 - 6x + 9\). This quadratic can be factored as follows:
\[ x^2 - 6x + 9 = (x - 3)(x - 3) = (x - 3)^2 \]
Now we can substitute this back into our factored expression:
\[ 4x^2(x^2 - 6x + 9) = 4x^2(x - 3)^2 \]
Thus, the complete factorization of the expression \(4x^4 - 24x^3 + 36x^2\) is:
\[ \boxed{4x^2(x - 3)^2} \]