Write as a single fraction. \[ \frac{2 x}{x^{2}+x-6}-\frac{2}{x+3}+\frac{5}{x+9} \] Simplify your answer as much as possible. Din

To solve the expression \[ \frac{2x}{x^{2} + x - 6} - \frac{2}{x + 3} + \frac{5}{x + 9} \] we first need to factor the quadratic in the first fraction. The expression \(x^2 + x - 6\) can be factored into \((x + 3)(x - 2)\). Thus, we rewrite the first fraction as: \[ \frac{2x}{(x + 3)(x - 2)} - \frac{2}{x + 3} + \frac{5}{x + 9} \] Next, we find a common denominator for all three fractions. The denominators are \((x + 3)(x - 2)\), \(x + 3\), and \(x + 9\). The least common denominator (LCD) is \((x + 3)(x - 2)(x + 9)\). Now we rewrite each fraction with the LCD: 1. For \(\frac{2x}{(x + 3)(x - 2)}\): \[ \frac{2x \cdot (x + 9)}{(x + 3)(x - 2)(x + 9)} = \frac{2x(x + 9)}{(x + 3)(x - 2)(x + 9)} = \frac{2x^2 + 18x}{(x + 3)(x - 2)(x + 9)} \] 2. For \(-\frac{2}{x + 3}\): \[ -\frac{2 \cdot (x - 2)(x + 9)}{(x + 3)(x - 2)(x + 9)} = -\frac{2(x^2 + 7x - 18)}{(x + 3)(x - 2)(x + 9)} = -\frac{2x^2 + 14x - 36}{(x + 3)(x - 2)(x + 9)} \] 3. For \(\frac{5}{x + 9}\): \[ \frac{5 \cdot (x + 3)(x - 2)}{(x + 3)(x - 2)(x + 9)} = \frac{5(x^2 + x - 6)}{(x + 3)(x - 2)(x + 9)} = \frac{5x^2 + 5x - 30}{(x + 3)(x - 2)(x + 9)} \] Now, we can combine all the fractions into a single fraction: \[ \frac{2x^2 + 18x - (2x^2 + 14x - 36) + (5x^2 + 5x - 30)}{(x + 3)(x - 2)(x + 9)} \] Simplifying the numerator: \[ 2x^2 + 18x - 2x^2 - 14x + 36 + 5x^2 + 5x - 30 \] Combine like terms: \[ (2x^2 - 2x^2 + 5x^2) + (18x - 14x + 5x) + (36 - 30) \] This becomes: \[ 5x^2 + 9x + 6 \] Thus, we have: \[ \frac{5x^2 + 9x + 6}{(x + 3)(x - 2)(x + 9)} \] Next, we can check if the numerator can be factored further. The quadratic \(5x^2 + 9x + 6\) can be factorized as: \[ (5x + 3)(x + 2) \] So we rewrite our expression: \[ \frac{(5x + 3)(x + 2)}{(x + 3)(x - 2)(x + 9)} \] The fully simplified form of the given fractions combined is: \[ \frac{(5x + 3)(x + 2)}{(x + 3)(x - 2)(x + 9)} \]