1. The problem is to simplify or factor the expression $4x^2 - 2x^3 - 6x$. 2. First, identify the greatest common factor (GCF) of all terms. The terms are $4x^2$, $-2x^3$, and $-6x$. The GCF is $2x$. 3. Factor out $2x$ from each term: $$4x^2 - 2x^3 - 6x = 2x(2x - x^2 - 3).$$ 4. Rewrite the expression inside the parentheses: $$2x - x^2 - 3 = -x^2 + 2x - 3 = -(x^2 - 2x + 3).$$ 5. Check if the quadratic $x^2 - 2x + 3$ can be factored further. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-2)^2 - 4 \times 1 \times 3 = 4 - 12 = -8.$$ Since the discriminant is negative, the quadratic does not factor over the real numbers. 6. Therefore, the fully factored form over the reals is: $$-2x(x^2 - 2x + 3).$$ 7. This is the simplified and factored form of the original expression.