Question

simplify the following expression completely.\\(\frac{x^2 + 6x - 7}{x^2 + 10x + 21}\\)

Explanation:

Step1: Factor numerator and denominator

Factor \(x^{2}+6x - 7\): We need two numbers that multiply to \(-7\) and add to \(6\). The numbers are \(7\) and \(-1\), so \(x^{2}+6x - 7=(x + 7)(x - 1)\).

Factor \(x^{2}+10x + 21\): We need two numbers that multiply to \(21\) and add to \(10\). The numbers are \(3\) and \(7\), so \(x^{2}+10x + 21=(x + 3)(x + 7)\).

So the expression becomes \(\frac{(x + 7)(x - 1)}{(x + 3)(x + 7)}\).

Step2: Cancel common factors

Cancel out the common factor \((x + 7)\) (assuming \(x
eq - 7\)): \(\frac{(x + 7)(x - 1)}{(x + 3)(x + 7)}=\frac{x - 1}{x + 3}\)

Answer:

\(\frac{x - 1}{x + 3}\)