1. **Problem Statement:** Simplify the rational function $ f(x) = \frac{x^2 + 3x - 40}{25 - x^2} $.\n\n2. **Formula and Rules:** To simplify rational functions, factor numerators and denominators completely, then cancel common factors. Remember to rewrite expressions in standard form and factor differences of squares when possible.\n\n3. **Step 1: Factor numerator and denominator.**\nNumerator: $x^2 + 3x - 40 = (x + 8)(x - 5)$\nDenominator: $25 - x^2 = (5)^2 - (x)^2 = (5 - x)(5 + x)$\n\n4. **Step 2: Put denominator in standard form and factor out $-1$ if needed.**\nRewrite denominator as $25 - x^2 = -(x^2 - 25) = -(x - 5)(x + 5)$\n\n5. **Step 3: Substitute back and simplify.**\n$$ f(x) = \frac{(x + 8)(x - 5)}{-(x - 5)(x + 5)} $$\n\n6. **Step 4: Cancel common factors $(x - 5)$.**\n$$ f(x) = \frac{(x + 8)\cancel{(x - 5)}}{-\cancel{(x - 5)}(x + 5)} = -\frac{x + 8}{x + 5} $$\n\n7. **Final simplified form:**\n$$ \boxed{f(x) = -\frac{x + 8}{x + 5}} $$\n\nThis simplification is valid for all $x \neq 5, -5$ because these values make the original denominator zero.