1. **State the problem:** Simplify the expression $$\frac{x^4 - 13x^2 + 36}{x^2 - 4}$$. 2. **Recall the formula and rules:** We will factor both numerator and denominator and then simplify by canceling common factors. 3. **Factor the denominator:** $$x^2 - 4 = (x - 2)(x + 2)$$ 4. **Factor the numerator:** Let $y = x^2$, then numerator becomes: $$y^2 - 13y + 36$$ We look for two numbers that multiply to 36 and add to -13: -9 and -4. So, $$y^2 - 13y + 36 = (y - 9)(y - 4)$$ Substitute back $y = x^2$: $$ (x^2 - 9)(x^2 - 4) $$ 5. **Rewrite the numerator:** $$ (x^2 - 9)(x^2 - 4) $$ 6. **Factor $x^2 - 9$ further:** $$x^2 - 9 = (x - 3)(x + 3)$$ 7. **Rewrite numerator fully factored:** $$ (x - 3)(x + 3)(x^2 - 4) $$ 8. **Rewrite the original expression:** $$\frac{(x - 3)(x + 3)(x^2 - 4)}{x^2 - 4}$$ 9. **Cancel common factor $x^2 - 4$:** $$\frac{(x - 3)(x + 3)\cancel{(x^2 - 4)}}{\cancel{x^2 - 4}} = (x - 3)(x + 3)$$ 10. **Final simplified form:** $$ (x - 3)(x + 3) = x^2 - 9 $$ **Answer:** $$x^2 - 9$$