Question

simplify the complex fraction.\\(
\frac{\left(\frac{x^2 + 5x - 14}{x + 6}\
ight)}{3x - 6}\\)
\\(\boxed{}\\), \\(x \
eq 2, -6\\)

Explanation:

Step1: Factor numerator and denominator

Factor \(x^2 + 5x - 14\): find two numbers that multiply to \(-14\) and add to \(5\), which are \(7\) and \(-2\). So \(x^2 + 5x - 14=(x + 7)(x - 2)\).
Factor \(3x - 6\): factor out \(3\), so \(3x - 6 = 3(x - 2)\).
The complex fraction becomes \(\frac{\frac{(x + 7)(x - 2)}{x + 6}}{3(x - 2)}\).

Step2: Divide by multiplying reciprocal

Dividing by a fraction is multiplying by its reciprocal. So \(\frac{(x + 7)(x - 2)}{x + 6}\times\frac{1}{3(x - 2)}\).

Step3: Cancel common factors

Cancel \((x - 2)\) from numerator and denominator (since \(x
eq2\), this is valid). We get \(\frac{x + 7}{3(x + 6)}\) or \(\frac{x + 7}{3x + 18}\).

Answer:

\(\frac{x + 7}{3x + 18}\) (or \(\frac{x + 7}{3(x + 6)}\))