1. **Problem:** Find the equation of the tangent line to $f(x) = x^5 - 5x + 1$ at $x = -2$. 2. **Formula:** The tangent line at $x=a$ is given by: $$y = f(a) + f'(a)(x - a)$$ where $f'(x)$ is the derivative of $f(x)$. 3. **Find the derivative:** $$f'(x) = 5x^4 - 5$$ 4. **Evaluate $f(-2)$:** $$f(-2) = (-2)^5 - 5(-2) + 1 = -32 + 10 + 1 = -21$$ 5. **Evaluate $f'(-2)$:** $$f'(-2) = 5(-2)^4 - 5 = 5(16) - 5 = 80 - 5 = 75$$ 6. **Write the tangent line equation:** $$y = f(-2) + f'(-2)(x - (-2)) = -21 + 75(x + 2)$$ 7. **Simplify:** $$y = -21 + 75x + 150 = 75x + 129$$ **Final answer:** $$\boxed{y = 75x + 129}$$