1. **State the problem:** Solve the equation $$\sqrt{5x + 3} = \sqrt{2x + 4} + 5$$. 2. **Isolate the square root terms:** We want to isolate one square root to eliminate them by squaring. 3. **Square both sides:** $$\left(\sqrt{5x + 3}\right)^2 = \left(\sqrt{2x + 4} + 5\right)^2$$ which simplifies to $$5x + 3 = (\sqrt{2x + 4})^2 + 2 \cdot 5 \cdot \sqrt{2x + 4} + 5^2$$ $$5x + 3 = 2x + 4 + 10\sqrt{2x + 4} + 25$$ 4. **Simplify the right side:** $$5x + 3 = 2x + 29 + 10\sqrt{2x + 4}$$ 5. **Isolate the square root term:** $$5x + 3 - 2x - 29 = 10\sqrt{2x + 4}$$ $$3x - 26 = 10\sqrt{2x + 4}$$ 6. **Square both sides again:** $$\left(3x - 26\right)^2 = \left(10\sqrt{2x + 4}\right)^2$$ $$\left(3x - 26\right)^2 = 100(2x + 4)$$ 7. **Expand and simplify:** $$9x^2 - 2 \cdot 3x \cdot 26 + 26^2 = 200x + 400$$ $$9x^2 - 156x + 676 = 200x + 400$$ 8. **Bring all terms to one side:** $$9x^2 - 156x + 676 - 200x - 400 = 0$$ $$9x^2 - 356x + 276 = 0$$ 9. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=9$, $b=-356$, $c=276$. Calculate discriminant: $$\Delta = (-356)^2 - 4 \cdot 9 \cdot 276 = 126736 - 9936 = 116800$$ Calculate roots: $$x = \frac{356 \pm \sqrt{116800}}{18}$$ $$\sqrt{116800} \approx 341.77$$ So, $$x_1 = \frac{356 + 341.77}{18} \approx \frac{697.77}{18} \approx 38.77$$ $$x_2 = \frac{356 - 341.77}{18} \approx \frac{14.23}{18} \approx 0.79$$ 10. **Check for extraneous solutions:** Plug $x_1 \approx 38.77$ into original equation: Left side: $$\sqrt{5(38.77) + 3} = \sqrt{193.85 + 3} = \sqrt{196.85} \approx 14.03$$ Right side: $$\sqrt{2(38.77) + 4} + 5 = \sqrt{77.54 + 4} + 5 = \sqrt{81.54} + 5 \approx 9.03 + 5 = 14.03$$ Matches, so $x_1$ is valid. Plug $x_2 \approx 0.79$ into original equation: Left side: $$\sqrt{5(0.79) + 3} = \sqrt{3.95 + 3} = \sqrt{6.95} \approx 2.64$$ Right side: $$\sqrt{2(0.79) + 4} + 5 = \sqrt{1.58 + 4} + 5 = \sqrt{5.58} + 5 \approx 2.36 + 5 = 7.36$$ Does not match, so $x_2$ is extraneous. **Final solution:** $$x \approx 38.77$$