To factor the polynomial \(20x^3 - 96x^2 - 144x\), you can start by factoring out the greatest common factor (GCF) from all the terms.
The GCF of the coefficients \(20\), \(-96\), and \(-144\) is \(4\), and all terms have at least one factor of \(x\). Thus, we can factor out \(4x\):
\[ 20x^3 - 96x^2 - 144x = 4x(5x^2 - 24x - 36) \]
Next, we need to factor the quadratic \(5x^2 - 24x - 36\). We can look for two numbers that multiply to \(5 \times -36 = -180\) and add to \(-24\). The numbers that meet these criteria are \(-30\) and \(6\).
Now we can split the middle term and factor by grouping:
\[ 5x^2 - 30x + 6x - 36 \]
Grouping gives us:
\[ (5x^2 - 30x) + (6x - 36) \]
Factoring each group results in:
\[ 5x(x - 6) + 6(x - 6) \]
Now we can factor out the common binomial factor \(x - 6\):
\[ (5x + 6)(x - 6) \]
Putting it all together, we have:
\[ 20x^3 - 96x^2 - 144x = 4x(5x + 6)(x - 6) \]
Thus, the correct linear factors of the polynomial \(20x^3 - 96x^2 - 144x\) are:
\[ \boxed{4x(5x + 6)(x - 6)} \]