Chapter 14 - A Preview of Calculus: The Limit, Derivative, and Integral of a Function - 14.4 The Tangent Problem; The Derivative - 14.4 Assess Your Understanding - Page 898: 24

$-4$

Factor each polynomial: $$f'(-1)\\=\lim_{x\to -1}\frac{f(x)-f(-1)}{x-(-1)}\\=\lim_{x\to -1}\frac{2x^2+1-(2(-1)^2+1))}{x+1}\\=\lim_{x\to -1}\frac{2x^2-2}{x+1}\\=\lim_{x\to -1}\frac{2(x+1)(x-1)}{(x+1)}.$$ Cancel the common factors: $$\lim_{x\to -1}2(x-1)\\=2(-1-1)=-4$$