1. **State the problem:** Simplify the expression $$\frac{x}{x-3} + \frac{2}{x^2-3}$$. 2. **Identify the denominators:** The denominators are $x-3$ and $x^2-3$. 3. **Factor the second denominator if possible:** $$x^2 - 3$$ cannot be factored further using integers, so it remains as is. 4. **Find a common denominator:** The common denominator is $$ (x-3)(x^2-3) $$. 5. **Rewrite each fraction with the common denominator:** $$\frac{x}{x-3} = \frac{x(x^2-3)}{(x-3)(x^2-3)}$$ $$\frac{2}{x^2-3} = \frac{2(x-3)}{(x-3)(x^2-3)}$$ 6. **Add the numerators:** $$\frac{x(x^2-3) + 2(x-3)}{(x-3)(x^2-3)}$$ 7. **Expand the numerators:** $$x(x^2-3) = x^3 - 3x$$ $$2(x-3) = 2x - 6$$ 8. **Combine the numerator terms:** $$x^3 - 3x + 2x - 6 = x^3 - x - 6$$ 9. **Final expression:** $$\frac{x^3 - x - 6}{(x-3)(x^2-3)}$$ 10. **Check if numerator can be factored:** Try to factor $x^3 - x - 6$ by rational root theorem. Test $x=2$: $$2^3 - 2 - 6 = 8 - 2 - 6 = 0$$ So, $x=2$ is a root. 11. **Divide numerator by $(x-2)$:** Using synthetic division: $$x^3 - x - 6 \div (x-2) = x^2 + 2x + 3$$ 12. **Rewrite numerator:** $$x^3 - x - 6 = (x-2)(x^2 + 2x + 3)$$ 13. **Final simplified form:** $$\frac{(x-2)(x^2 + 2x + 3)}{(x-3)(x^2-3)}$$ This is the simplified form of the original expression.