Question

(\frac{x}{x + 2} + \frac{x - 12}{x^2 - 3x - 16})

Explanation:

Step1: Factor the denominator

Factor \(x^2 - 3x - 10\). We need two numbers that multiply to \(-10\) and add to \(-3\). The numbers are \(-5\) and \(2\). So, \(x^2 - 3x - 10=(x - 5)(x + 2)\).

Step2: Rewrite the expression

The original expression is \(\frac{x}{x + 2}+\frac{x - 12}{x^2 - 3x - 10}\). Substituting the factored form of the denominator, we get \(\frac{x}{x + 2}+\frac{x - 12}{(x - 5)(x + 2)}\).

Step3: Find a common denominator

The common denominator is \((x - 5)(x + 2)\). Rewrite the first fraction with the common denominator: \(\frac{x(x - 5)}{(x - 5)(x + 2)}+\frac{x - 12}{(x - 5)(x + 2)}\).

Step4: Combine the fractions

Combine the numerators over the common denominator: \(\frac{x(x - 5)+x - 12}{(x - 5)(x + 2)}\).

Step5: Simplify the numerator

Expand \(x(x - 5)\): \(x^2 - 5x\). Then add \(x - 12\): \(x^2 - 5x+x - 12=x^2 - 4x - 12\).

Step6: Factor the numerator

Factor \(x^2 - 4x - 12\). We need two numbers that multiply to \(-12\) and add to \(-4\). The numbers are \(-6\) and \(2\). So, \(x^2 - 4x - 12=(x - 6)(x + 2)\).

Step7: Simplify the fraction

The expression becomes \(\frac{(x - 6)(x + 2)}{(x - 5)(x + 2)}\). Cancel out the common factor \((x + 2)\) (assuming \(x
eq - 2\)).

Answer:

\(\frac{x - 6}{x - 5}\) (for \(x
eq - 2,5\))