1. **State the problem:** Solve for $x$ in the equation $$\sqrt{5x+4} - \sqrt{x} = 4.$$\n\n2. **Isolate one square root:** Add $\sqrt{x}$ to both sides to get $$\sqrt{5x+4} = 4 + \sqrt{x}.$$\n\n3. **Square both sides:** To eliminate the square root on the left, square both sides: $$\left(\sqrt{5x+4}\right)^2 = \left(4 + \sqrt{x}\right)^2.$$\nThis gives $$5x + 4 = 16 + 8\sqrt{x} + x.$$\n\n4. **Rearrange terms:** Move all terms except the square root to one side: $$5x + 4 - 16 - x = 8\sqrt{x}$$ which simplifies to $$4x - 12 = 8\sqrt{x}.$$\n\n5. **Isolate the square root:** Divide both sides by 8: $$\frac{4x - 12}{8} = \sqrt{x}.$$\nSimplify the fraction: $$\sqrt{x} = \frac{\cancel{4}(x - 3)}{\cancel{8}2} = \frac{x - 3}{2}.$$\n\n6. **Square both sides again:** To eliminate the square root, square both sides: $$\left(\sqrt{x}\right)^2 = \left(\frac{x - 3}{2}\right)^2,$$ which gives $$x = \frac{(x - 3)^2}{4}.$$\n\n7. **Multiply both sides by 4:** $$4x = (x - 3)^2 = x^2 - 6x + 9.$$\n\n8. **Bring all terms to one side:** $$0 = x^2 - 6x + 9 - 4x,$$ which simplifies to $$0 = x^2 - 10x + 9.$$\n\n9. **Solve quadratic equation:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ with $a=1$, $b=-10$, $c=9$: $$x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 9}}{2} = \frac{10 \pm \sqrt{100 - 36}}{2} = \frac{10 \pm \sqrt{64}}{2}.$$\n\n10. **Calculate roots:** $$x = \frac{10 \pm 8}{2}.$$\nSo, $$x_1 = \frac{10 + 8}{2} = 9,$$ and $$x_2 = \frac{10 - 8}{2} = 1.$$\n\n11. **Check for extraneous solutions:** Substitute $x=9$ into the original equation: $$\sqrt{5(9)+4} - \sqrt{9} = \sqrt{45+4} - 3 = \sqrt{49} - 3 = 7 - 3 = 4,$$ which is true.\nSubstitute $x=1$: $$\sqrt{5(1)+4} - \sqrt{1} = \sqrt{5+4} - 1 = \sqrt{9} - 1 = 3 - 1 = 2,$$ which is false.\n\n**Final answer:** $$\boxed{9}.$$