Answer
$-24$
Work Step by Step
Factor each polynomial:
$$f'(3)\\=\lim_{x\to 3}\frac{f(x)-f(3)}{x-3}\\=\lim_{x\to 3}\frac{-4x^2+5-(-4(3)^2+5))}{x-3}\\=\lim_{x\to 3}\frac{-4x^2+36}{x-3}\\=\lim_{x\to 3}\frac{-4(x^2-9)}{x-3}\\=\lim_{x\to 3}\frac{-4(x+3)(x-3)}{(x-3)}.$$
Cancel the common factors: $$\lim_{x\to 3}-4(x+3)\\=-4(3+3)=-24$$