1. **Problem statement:** Simplify and prove that the expression $$A = \left( \frac{x - 2}{x^3 - 3x^2} - \frac{6}{x^4 - 9x^2} + \frac{x + 2}{x^2 + 3x^2} \right) \cdot \frac{4042}{x^2}$$ does not depend on $x$ for values where it is defined. 2. **Step 1: Simplify each denominator and numerator where possible.** - Note that $x^3 - 3x^2 = x^2(x - 3)$. - Also, $x^4 - 9x^2 = x^2(x^2 - 9) = x^2(x - 3)(x + 3)$. - And $x^2 + 3x^2 = 4x^2$. 3. **Rewrite the expression:** $$A = \left( \frac{x - 2}{x^2(x - 3)} - \frac{6}{x^2(x - 3)(x + 3)} + \frac{x + 2}{4x^2} \right) \cdot \frac{4042}{x^2}$$ 4. **Find a common denominator inside the parentheses:** The common denominator is $4x^2(x - 3)(x + 3)$. Rewrite each term with this denominator: - First term numerator multiplied by $4(x + 3)$: $$\frac{(x - 2) \cdot 4(x + 3)}{4x^2(x - 3)(x + 3)}$$ - Second term numerator multiplied by 4: $$\frac{6 \cdot 4}{4x^2(x - 3)(x + 3)} = \frac{24}{4x^2(x - 3)(x + 3)}$$ - Third term numerator multiplied by $(x - 3)(x + 3)$: $$\frac{(x + 2)(x - 3)(x + 3)}{4x^2(x - 3)(x + 3)}$$ 5. **Combine the numerators:** $$\frac{4(x - 2)(x + 3) - 24 + (x + 2)(x - 3)(x + 3)}{4x^2(x - 3)(x + 3)}$$ 6. **Expand and simplify numerator:** - Expand $4(x - 2)(x + 3)$: $$4(x^2 + 3x - 2x - 6) = 4(x^2 + x - 6) = 4x^2 + 4x - 24$$ - Expand $(x + 2)(x - 3)(x + 3)$: First, $(x - 3)(x + 3) = x^2 - 9$. Then, $(x + 2)(x^2 - 9) = x^3 + 2x^2 - 9x - 18$. 7. **Sum all parts:** $$4x^2 + 4x - 24 - 24 + x^3 + 2x^2 - 9x - 18 = x^3 + (4x^2 + 2x^2) + (4x - 9x) + (-24 - 24 - 18)$$ $$= x^3 + 6x^2 - 5x - 66$$ 8. **So the expression inside parentheses is:** $$\frac{x^3 + 6x^2 - 5x - 66}{4x^2(x - 3)(x + 3)}$$ 9. **Multiply by $\frac{4042}{x^2}$:** $$A = \frac{4042}{x^2} \cdot \frac{x^3 + 6x^2 - 5x - 66}{4x^2(x - 3)(x + 3)} = \frac{4042(x^3 + 6x^2 - 5x - 66)}{4x^4(x - 3)(x + 3)}$$ 10. **Factor numerator $x^3 + 6x^2 - 5x - 66$ by grouping:** Group as $(x^3 + 6x^2) + (-5x - 66)$ $$= x^2(x + 6) - 11(5x + 6)$$ Try to factor further or test roots. 11. **Test $x=3$:** $$3^3 + 6(3)^2 - 5(3) - 66 = 27 + 54 - 15 - 66 = 0$$ So, $(x - 3)$ is a factor. 12. **Divide numerator by $(x - 3)$:** Using polynomial division or synthetic division: $$x^3 + 6x^2 - 5x - 66 = (x - 3)(x^2 + 9x + 22)$$ 13. **Rewrite $A$:** $$A = \frac{4042 (x - 3)(x^2 + 9x + 22)}{4 x^4 (x - 3)(x + 3)}$$ 14. **Cancel $(x - 3)$:** $$A = \frac{4042 (x^2 + 9x + 22)}{4 x^4 (x + 3)}$$ 15. **No further obvious simplification; check if $A$ depends on $x$.** Since the problem states $A$ does not depend on $x$, check for possible simplification or error. 16. **Re-examine original problem or consider possible typo in problem statement.** Since the problem is complex and the user asked to solve only the first question, we provide the above detailed steps. **Final answer:** $$A = \frac{4042 (x^2 + 9x + 22)}{4 x^4 (x + 3)}$$ which depends on $x$ unless further conditions are given. --- **Slug:** expression independence **Subject:** algebra **Desmos:** {"latex":"y=\frac{4042 (x^2 + 9x + 22)}{4 x^4 (x + 3)}","features":{"intercepts":true,"extrema":true}} **q_count:** 5