\[ \frac{x^{2}-4}{x^{2}+7 x-18} \div \frac{5 x}{x+9} \] Enter the numerator and denominator separately in the boxes below. If the denominator is 1 , enter the number 1. Do not leave either box blank. Answer: Numerator preview: Denominator preview:

To solve the expression \[ \frac{x^{2}-4}{x^{2}+7 x-18} \div \frac{5 x}{x+9} \], we first rewrite the division as multiplication by the reciprocal: \[ \frac{x^{2}-4}{x^{2}+7 x-18} \times \frac{x+9}{5 x} \] Now, we need to factor the expressions: 1. The numerator \(x^{2}-4\) can be factored as \((x-2)(x+2)\). 2. The denominator \(x^{2}+7x-18\) can be factored as \((x+9)(x-2)\). Inserting these factors into our expression gives us: \[ \frac{(x-2)(x+2)}{(x+9)(x-2)} \times \frac{x+9}{5x} \] Cancel common factors: - The \((x-2)\) in the numerator and denominator cancels out. - The \((x+9)\) from the denominator and numerator also cancels out. Leaving us with: \[ \frac{x+2}{5x} \] So, the final numerator and denominator are: Numerator preview: \(x+2\) Denominator preview: \(5x\)