1. **State the problem:** Find the critical points of the function $$f(x) = \frac{3x^2 + 5x + 25}{x + 2}$$. 2. **Recall the formula:** Critical points occur where the derivative $$f'(x)$$ is zero or undefined. 3. **Find the derivative using the quotient rule:** $$f'(x) = \frac{(6x + 5)(x + 2) - (3x^2 + 5x + 25)(1)}{(x + 2)^2}$$ 4. **Simplify the numerator:** $$ (6x + 5)(x + 2) - (3x^2 + 5x + 25) = (6x^2 + 12x + 5x + 10) - (3x^2 + 5x + 25) $$ $$= 6x^2 + 17x + 10 - 3x^2 - 5x - 25 = 3x^2 + 12x - 15$$ 5. **Set the numerator equal to zero to find critical points:** $$3x^2 + 12x - 15 = 0$$ 6. **Divide entire equation by 3:** $$\cancel{3}x^2 + \cancel{3} \cdot 4 x - \cancel{3} \cdot 5 = 0 \Rightarrow x^2 + 4x - 5 = 0$$ 7. **Factor the quadratic:** $$ (x + 5)(x - 1) = 0 $$ 8. **Solve for x:** $$ x = -5 \quad \text{or} \quad x = 1 $$ 9. **Check where derivative is undefined:** Derivative is undefined where denominator $$ (x + 2)^2 = 0 \Rightarrow x = -2 $$, which is not in the domain of $$f(x)$$. 10. **Conclusion:** The critical points of $$f(x)$$ are at $$x = -5$$ and $$x = 1$$.