QuestionDecember 12, 2025

22 (x+3)/(x-1)+(x-4)/(x+2)

22 (x+3)/(x-1)+(x-4)/(x+2)
22 (x+3)/(x-1)+(x-4)/(x+2)

Solution
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Answer

22\frac{x+3}{x-1}+\frac{x-4}{x+2} = \frac{2x^2 + 10}{(x-1)(x+2)} Explanation 1. Find a common denominator The common denominator is (x-1)(x+2). 2. Rewrite each fraction with the common denominator \frac{x+3}{x-1} = \frac{(x+3)(x+2)}{(x-1)(x+2)}, \frac{x-4}{x+2} = \frac{(x-4)(x-1)}{(x-1)(x+2)} 3. Add the numerators (x+3)(x+2) + (x-4)(x-1) over (x-1)(x+2). 4. Expand the numerators (x+3)(x+2) = x^2 + 5x + 6, (x-4)(x-1) = x^2 - 5x + 4. 5. Combine like terms [x^2 + 5x + 6] + [x^2 - 5x + 4] = 2x^2 + 10. 6. Write the final simplified expression \frac{2x^2 + 10}{(x-1)(x+2)}.

Explanation

1. Find a common denominator<br /> The common denominator is $(x-1)(x+2)$.<br />2. Rewrite each fraction with the common denominator<br /> $\frac{x+3}{x-1} = \frac{(x+3)(x+2)}{(x-1)(x+2)}$, $\frac{x-4}{x+2} = \frac{(x-4)(x-1)}{(x-1)(x+2)}$<br />3. Add the numerators<br /> $(x+3)(x+2) + (x-4)(x-1)$ over $(x-1)(x+2)$.<br />4. Expand the numerators<br /> $(x+3)(x+2) = x^2 + 5x + 6$, $(x-4)(x-1) = x^2 - 5x + 4$.<br />5. Combine like terms<br /> $[x^2 + 5x + 6] + [x^2 - 5x + 4] = 2x^2 + 10$.<br />6. Write the final simplified expression<br /> $\frac{2x^2 + 10}{(x-1)(x+2)}$.
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