Answer
$2x ^3 - 3x^2 - 11x + 6 = (x-3)(x+2)(x-\frac{1}{2})$
Work Step by Step
By using long division, the exercise already gives us one of the factors of $2x^3 - 3x^2 - 11x + 6$ such that we can write it as follows: $$(x - 3)(2x^2 + 3x - 2)$$ To further factor the equation, we can now use the Quadratic Formula, since the remaining polynomial is a quadratic polynomial: $$x = \frac{-3 \frac{+}{} \sqrt{3^2 - 4(2)(-2)}}{2(2)}$$ $$x = \frac{-3 \frac{+}{} \sqrt{25}}{4}$$ $$x = \frac{-3 \frac{+}{} 5}{4}$$ which means that the remaining factors are $$(x - \frac{-3 - 5}{4})$$ $$(x + 2)$$ and $$(x - \frac{-3 + 5}{4})$$ $$(x - \frac{1}{2})$$
