1. **State the problem:** Find the limit $$\lim_{x \to 5} \frac{-3x^2 + 75}{x - 5}$$. 2. **Identify the issue:** Direct substitution of $x=5$ gives $$\frac{-3(5)^2 + 75}{5 - 5} = \frac{-3 \cdot 25 + 75}{0} = \frac{-75 + 75}{0} = \frac{0}{0}$$ which is an indeterminate form. 3. **Use algebraic simplification:** Factor the numerator: $$-3x^2 + 75 = -3(x^2 - 25) = -3(x - 5)(x + 5)$$ 4. **Rewrite the limit expression:** $$\lim_{x \to 5} \frac{-3(x - 5)(x + 5)}{x - 5}$$ 5. **Cancel the common factor $(x - 5)$:** $$\lim_{x \to 5} \frac{-3\cancel{(x - 5)}(x + 5)}{\cancel{(x - 5)}} = \lim_{x \to 5} -3(x + 5)$$ 6. **Evaluate the simplified limit:** $$-3(5 + 5) = -3 \times 10 = -30$$ **Final answer:** $$\boxed{-30}$$