To factor out the greatest common factor (GCF) from the expression \(6x^3 + 6x^2 - 120x\), we first need to identify the GCF of the coefficients and the variable terms.
The coefficients are \(6\), \(6\), and \(-120\). The GCF of these numbers is \(6\).
Each term contains at least one \(x\), so we can factor out the smallest power of \(x\), which is \(x\). Therefore, the GCF of the entire expression is \(6x\).
Now we can factor \(6x\) out of the expression:
\[ 6x^3 + 6x^2 - 120x = 6x(x^2 + x - 20) \]
Next, we need to factor the quadratic \(x^2 + x - 20\). We look for two numbers that multiply to \(-20\) (the constant term) and add to \(1\) (the coefficient of the linear term).
The numbers that meet these criteria are \(5\) and \(-4\):
\[ x^2 + x - 20 = (x + 5)(x - 4) \]
Now we can complete the factorization of the original expression:
\[ 6x^3 + 6x^2 - 120x = 6x(x + 5)(x - 4) \]
Thus, the fully factored form of the expression is:
\[ \boxed{6x(x + 5)(x - 4)} \]