Answer
$20$
Work Step by Step
Factor each polynomial:
$$f'(2)\\=\lim_{x\to 2}\frac{f(x)-f(2)}{x-2}\\=\lim_{x\to 2}\frac{2x^3-x^2-(2(2)^3-(2)^2)}{x-2}\\=\lim_{x\to 2}\frac{2x^3-x^2-12}{x-2}\\=\lim_{x\to 2}\frac{(x-2)(2x^2+3x+6)}{(x-2)}.$$
Cancel the common factors: $$\lim_{x\to 2}2x^2+3x+6\\=2(2)^2+3(2)+6=20$$