9) \( \frac{5}{x^{2}-x-6}+\frac{3 x}{3-x}+\frac{4}{2+x} \)

Simplify the expression by following steps: - step0: Solution: \(\left(\frac{5}{\left(x^{2}-x-6\right)}\right)+\left(\frac{3x}{\left(3-x\right)}\right)+\left(\frac{4}{\left(2+x\right)}\right)\) - step1: Remove the parentheses: \(\left(\frac{5}{x^{2}-x-6}\right)+\left(\frac{3x}{3-x}\right)+\left(\frac{4}{2+x}\right)\) - step2: Remove the parentheses: \(\frac{5}{x^{2}-x-6}+\left(\frac{3x}{3-x}\right)+\left(\frac{4}{2+x}\right)\) - step3: Remove the parentheses: \(\frac{5}{x^{2}-x-6}+\frac{3x}{3-x}+\left(\frac{4}{2+x}\right)\) - step4: Remove the parentheses: \(\frac{5}{x^{2}-x-6}+\frac{3x}{3-x}+\frac{4}{2+x}\) - step5: Rewrite the fractions: \(\frac{5}{x^{2}-x-6}-\frac{3x}{-3+x}+\frac{4}{2+x}\) - step6: Factor the expression: \(\frac{5}{\left(x+2\right)\left(x-3\right)}-\frac{3x}{-3+x}+\frac{4}{2+x}\) - step7: Reduce fractions to a common denominator: \(\frac{5}{\left(x+2\right)\left(x-3\right)}-\frac{3x\left(x+2\right)}{\left(-3+x\right)\left(x+2\right)}+\frac{4\left(x-3\right)}{\left(2+x\right)\left(x-3\right)}\) - step8: Rewrite the expression: \(\frac{5}{\left(x+2\right)\left(x-3\right)}-\frac{3x\left(x+2\right)}{\left(x+2\right)\left(x-3\right)}+\frac{4\left(x-3\right)}{\left(x+2\right)\left(x-3\right)}\) - step9: Transform the expression: \(\frac{5-3x\left(x+2\right)+4\left(x-3\right)}{\left(x+2\right)\left(x-3\right)}\) - step10: Multiply the terms: \(\frac{5-\left(3x^{2}+6x\right)+4\left(x-3\right)}{\left(x+2\right)\left(x-3\right)}\) - step11: Multiply the terms: \(\frac{5-\left(3x^{2}+6x\right)+4x-12}{\left(x+2\right)\left(x-3\right)}\) - step12: Calculate: \(\frac{-7-3x^{2}-2x}{\left(x+2\right)\left(x-3\right)}\) - step13: Rewrite the fraction: \(-\frac{7+3x^{2}+2x}{\left(x+2\right)\left(x-3\right)}\) - step14: Multiply the terms: \(-\frac{7+3x^{2}+2x}{x^{2}-x-6}\) To simplify the expression \[ \frac{5}{x^{2}-x-6}+\frac{3 x}{3-x}+\frac{4}{2+x}, \] we can follow these steps: 1. **Factor the Denominators**: - The first term has a denominator \(x^2 - x - 6\). We can factor this as \((x-3)(x+2)\). - The second term has a denominator \(3 - x\), which can be rewritten as \(-(x - 3)\). - The third term has a denominator \(2 + x\), which can be rewritten as \((x + 2)\). 2. **Rewrite the Expression**: The expression can be rewritten as: \[ \frac{5}{(x-3)(x+2)} - \frac{3x}{(x-3)} + \frac{4}{(x+2)}. \] 3. **Combine the Terms**: To combine these fractions, we need a common denominator, which is \((x-3)(x+2)\). We rewrite each term: - The first term remains \(\frac{5}{(x-3)(x+2)}\). - The second term becomes \(-\frac{3x(x+2)}{(x-3)(x+2)}\). - The third term becomes \(\frac{4(x-3)}{(x-3)(x+2)}\). 4. **Combine the Numerators**: Now we can combine the numerators: \[ \frac{5 - 3x(x+2) + 4(x-3)}{(x-3)(x+2)}. \] 5. **Simplify the Numerator**: Expanding the numerator: - \( -3x(x+2) = -3x^2 - 6x \) - \( 4(x-3) = 4x - 12 \) Combining these gives: \[ 5 - 3x^2 - 6x + 4x - 12 = -3x^2 - 2x - 7. \] 6. **Final Expression**: Thus, the simplified expression is: \[ -\frac{3x^2 + 2x + 7}{(x-3)(x+2)}. \] In conclusion, the simplified form of the original expression is: \[ -\frac{3x^2 + 2x + 7}{(x-3)(x+2)}. \]