1. **State the problem:** Find all x-intercepts of the function $$f(x) = \frac{x^2 + 8x}{x^2 + 4x - 21}$$. 2. **Recall the rule for x-intercepts:** The x-intercepts occur where the function equals zero, i.e., where the numerator is zero and the denominator is not zero. 3. **Set the numerator equal to zero:** $$x^2 + 8x = 0$$ 4. **Factor the numerator:** $$x(x + 8) = 0$$ 5. **Solve for x:** $$x = 0 \quad \text{or} \quad x = -8$$ 6. **Check the denominator at these x-values to avoid division by zero:** $$x^2 + 4x - 21 = (x + 7)(x - 3)$$ - At $$x=0$$: $$0 + 0 - 21 = -21 \neq 0$$, so valid. - At $$x=-8$$: $$64 - 32 - 21 = 11 \neq 0$$, so valid. 7. **Conclusion:** Both $$x=0$$ and $$x=-8$$ are x-intercepts. However, the user states there is only one x-intercept, so let's verify if any simplification cancels factors. 8. **Factor numerator and denominator fully:** - Numerator: $$x(x+8)$$ - Denominator: $$(x+7)(x-3)$$ No common factors to cancel. 9. **Therefore, both points are valid x-intercepts:** $$ (0,0) \quad \text{and} \quad (-8,0) $$ Since the user says there is only one x-intercept, the correct answer is both points, but if forced to choose one, it might be a misunderstanding. **Final answer:** The x-intercepts are $$\boxed{(0,0) \text{ and } (-8,0)}$$.