Divide. (Simplify your answer completely.) \[ \frac{x^{2}+3 x-18}{x^{2}+2 x-15} \div \frac{x^{2}+2 x-24}{x^{2}+3 x-4} \]

To simplify the expression, we will first rewrite the division as multiplication by the reciprocal and then factor the polynomials involved. 1. Start with the original expression: \[ \frac{x^{2}+3 x-18}{x^{2}+2 x-15} \div \frac{x^{2}+2 x-24}{x^{2}+3 x-4} \] This can be rewritten as: \[ \frac{x^{2}+3 x-18}{x^{2}+2 x-15} \times \frac{x^{2}+3 x-4}{x^{2}+2 x-24} \] 2. Next, we will factor each polynomial: - For \(x^{2}+3x-18\), we look for two numbers that multiply to \(-18\) and add to \(3\). These numbers are \(6\) and \(-3\). Thus: \[ x^{2}+3x-18 = (x+6)(x-3) \] - For \(x^{2}+2x-15\), we need two numbers that multiply to \(-15\) and add to \(2\). They are \(5\) and \(-3\). Thus: \[ x^{2}+2x-15 = (x+5)(x-3) \] - For \(x^{2}+2x-24\), we look for two numbers that multiply to \(-24\) and add to \(2\). They are \(6\) and \(-4\). Thus: \[ x^{2}+2x-24 = (x+6)(x-4) \] - For \(x^{2}+3x-4\), we look for two numbers that multiply to \(-4\) and add to \(3\). They are \(4\) and \(-1\). Thus: \[ x^{2}+3x-4 = (x+4)(x-1) \] 3. Now we can rewrite the entire expression with the factored forms: \[ \frac{(x+6)(x-3)}{(x+5)(x-3)} \times \frac{(x+4)(x-1)}{(x+6)(x-4)} \] 4. Next, we can cancel out any common factors: - \(x+6\) in the numerator and denominator. - \(x-3\) in the numerator and denominator. This simplifies our expression to: \[ \frac{(x+4)(x-1)}{(x+5)(x-4)} \] 5. Thus, the final answer is: \[ \frac{(x+4)(x-1)}{(x+5)(x-4)} \] Now, the expression is simplified completely.