1. Stating the problem: Solve the equation $$\sqrt{3}a - 5 + \sqrt{2}a + 3 + 1 = 0$$ for $a$. 2. Combine like terms: Group the terms involving $a$ and the constants separately. $$\sqrt{3}a + \sqrt{2}a + (-5 + 3 + 1) = 0$$ 3. Simplify the constants: $$-5 + 3 + 1 = -1$$ So the equation becomes: $$\sqrt{3}a + \sqrt{2}a - 1 = 0$$ 4. Factor out $a$: $$a(\sqrt{3} + \sqrt{2}) - 1 = 0$$ 5. Isolate $a$: $$a(\sqrt{3} + \sqrt{2}) = 1$$ 6. Solve for $a$: $$a = \frac{1}{\sqrt{3} + \sqrt{2}}$$ 7. Rationalize the denominator: Multiply numerator and denominator by the conjugate $\sqrt{3} - \sqrt{2}$: $$a = \frac{1}{\sqrt{3} + \sqrt{2}} \times \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3})^2 - (\sqrt{2})^2}$$ 8. Simplify the denominator: $$(\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1$$ 9. Final answer: $$a = \sqrt{3} - \sqrt{2}$$ This is the solution for $a$ in the given equation.