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: 27

$7$

Factor each polynomial: $$f'(-1)\\=\lim_{x\to -1}\frac{f(x)-f(-1)}{x-(-1)}\\=\lim_{x\to -1}\frac{x^3+4x-((-1)^3+4(-1))}{x+1}\\=\lim_{x\to -1}\frac{x^3+4x+5}{x+1}\\=\lim_{x\to -1}\frac{(x+1)(x^2-x+5)}{(x+1)}.$$ Cancel the common factors: $$\lim_{x\to -1}x^2-x+5\\=(-1)^2-(-1)+5=7$$