Question

simplify.
\\(\frac{\frac{x^2 - 3x - 10}{x + 3}}{\frac{2x^2 - 7x - 15}{x^2 - 9}}\\)
\\(\circ\\) \\(\frac{x^2 + x - 6}{2x + 3}\\)
\\(\circ\\) \\(\frac{x^2 - 5x - 6}{2x - 3}\\)
\\(\circ\\) \\(\frac{x^2 - x - 6}{2x + 3}\\)
\\(\circ\\) \\(\frac{x^2 - 5x + 6}{2x - 3}\\)

Explanation:

Step1: Simplify the numerator's fraction

First, factor the quadratic in the numerator of the complex fraction: \(x^2 - 3x - 10=(x - 5)(x + 2)\). So the top fraction is \(\frac{(x - 5)(x + 2)}{x + 3}\).

Step2: Simplify the denominator's fraction

Factor the quadratics in the denominator's fraction. For \(2x^2 - 7x - 15\), we get \((2x + 3)(x - 5)\). For \(x^2 - 9\), it's a difference of squares: \((x + 3)(x - 3)\). So the bottom fraction is \(\frac{(2x + 3)(x - 5)}{(x + 3)(x - 3)}\).

Step3: Divide the two fractions

Dividing by a fraction is multiplying by its reciprocal. So we have \(\frac{(x - 5)(x + 2)}{x + 3}\div\frac{(2x + 3)(x - 5)}{(x + 3)(x - 3)}=\frac{(x - 5)(x + 2)}{x + 3}\times\frac{(x + 3)(x - 3)}{(2x + 3)(x - 5)}\).

Step4: Cancel common factors

Cancel out \((x - 5)\) and \((x + 3)\) from numerator and denominator. We are left with \(\frac{(x + 2)(x - 3)}{2x + 3}\).

Step5: Expand the numerator

Expand \((x + 2)(x - 3)=x^2 - 3x + 2x - 6=x^2 - x - 6\). So the simplified form is \(\frac{x^2 - x - 6}{2x + 3}\).

Answer:

\(\frac{x^2 - x - 6}{2x + 3}\) (corresponding to the third option, e.g., if the third option is \(\boldsymbol{\frac{x^2 - x - 6}{2x + 3}}\), then the answer is that option)