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: 30

$8$

Factor each polynomial: $$f'(-1)\\=\lim_{x\to -1}\frac{f(x)-f(1)}{x-(-1)}\\=\lim_{x\to -1}\frac{x^3-2x^2+x-((-1)^3-2(-1)^2+(-1))}{x+1}\\=\lim_{x\to -1}\frac{x^3-2x^2+x+4}{x+1}\\=\lim_{x\to -1}\frac{((x+1)(x^2-3x+4)}{(x+1)}.$$ Cancel the common factors: $$\lim_{x\to -1}x^2-3x+4\\=(-1)^2-3(-1)+4=8$$