To factor the expression \(12x^3 - 44x^2 - 80x\), you can start by factoring out the greatest common factor (GCF) from all the terms. The GCF of the coefficients 12, -44, and -80 is 4, and the GCF of the variables \(x^3\), \(x^2\), and \(x\) is \(x\). So, factor out 4x:
\[4x(3x^2 - 11x - 20)\]
Now, you have the expression factored with the GCF factored out. Next, let's factor the quadratic expression \(3x^2 - 11x - 20\):
To factor the quadratic expression \(3x^2 - 11x - 20\), you can look for two numbers that multiply to the constant term (-20) and add up to the coefficient of the middle term (-11). Those numbers are -15 and +4. So, you can factor the quadratic as follows:
\[3x^2 - 11x - 20 = (3x + 4)(x - 5)\]
Now, you have factored the quadratic expression. Combine it with the GCF factored out earlier:
\[4x(3x + 4)(x - 5)\]
So, the factored form of \(12x^3 - 44x^2 - 80x\) is \(4x(3x + 4)(x - 5)\).