1. **State the problem:** Solve the equation $x + 5 = \sqrt{5x + 21}$ for $x$. 2. **Recall the formula and rules:** To solve equations involving square roots, isolate the square root term and then square both sides to eliminate the root. Remember to check for extraneous solutions after squaring. 3. **Isolate and square:** $$x + 5 = \sqrt{5x + 21}$$ Square both sides: $$(x + 5)^2 = (\sqrt{5x + 21})^2$$ $$x^2 + 10x + 25 = 5x + 21$$ 4. **Simplify and rearrange:** $$x^2 + 10x + 25 - 5x - 21 = 0$$ $$x^2 + 5x + 4 = 0$$ 5. **Factor the quadratic:** $$(x + 4)(x + 1) = 0$$ So, $x = -4$ or $x = -1$. 6. **Check for extraneous solutions:** - For $x = -4$: $$-4 + 5 = 1$$ $$\sqrt{5(-4) + 21} = \sqrt{-20 + 21} = \sqrt{1} = 1$$ Left side equals right side, so $x = -4$ is valid. - For $x = -1$: $$-1 + 5 = 4$$ $$\sqrt{5(-1) + 21} = \sqrt{-5 + 21} = \sqrt{16} = 4$$ Left side equals right side, so $x = -1$ is valid. **Final answer:** $-4, -1$