Answer
$8$
Work Step by Step
Factor each polynomial:
$$f'(2)\\=\lim_{x\to 2}\frac{f(x)-f(2)}{x-2}\\=\lim_{x\to 2}\frac{3x^2-4x-(3(2)^2-4(2))}{x-2}\\=\lim_{x\to 2}\frac{3x^2-4x-4}{x-2}\\=\lim_{x\to 2}\frac{(x-2)(3x+2)}{x-2}.$$
Cancel the common factors: $$\lim_{x\to 2}(3x+2)\\=3(2)+2=8$$