The problem requires us to factor the expression $12x^3 - 5x^2 - 3x$.

1. Problem Description

The problem requires us to factor the expression $12x^3 - 5x^2 - 3x$.

2. Solution Steps

First, we observe that the variable $x$ is a common factor in all three terms of the expression. Therefore, we can factor out an $x$.

$12x^3 - 5x^2 - 3x = x(12x^2 - 5x - 3)$

Now, we need to factor the quadratic expression $12x^2 - 5x - 3$. We are looking for two binomials of the form $(ax + b)(cx + d)$ such that $ac = 12$, $ad + bc = -5$, and $bd = -3$.

We can try different combinations of factors for 12 and -

3. Let's try $a=3$, $c=4$. Then we have $(3x+b)(4x+d)$. We need $3d + 4b = -5$ and $bd = -3$.

Let's try $b=1$ and $d=-3$. Then $3(-3) + 4(1) = -9 + 4 = -5$. This works!

So, we have $(3x+1)(4x-3)$.

Therefore, $12x^2 - 5x - 3 = (3x+1)(4x-3)$.

Finally, substitute this back into the original expression.

$x(12x^2 - 5x - 3) = x(3x+1)(4x-3)$

3. Final Answer

The factored expression is $x(3x+1)(4x-3)$.