Question

1. \\(\frac{x^2 - x - 12}{5x^2 + 14x - 3}\\)

Explanation:

Step1: Factor numerator

Factor \(x^2 - x - 12\). We need two numbers that multiply to \(-12\) and add to \(-1\). Those numbers are \(-4\) and \(3\). So, \(x^2 - x - 12=(x - 4)(x + 3)\).

Step2: Factor denominator

Factor \(5x^2 + 14x - 3\). We need two numbers that multiply to \(5\times(-3)=-15\) and add to \(14\). Those numbers are \(15\) and \(-1\). Rewrite the middle term: \(5x^2 + 15x - x - 3\). Group and factor: \(5x(x + 3)-1(x + 3)=(5x - 1)(x + 3)\).

Step3: Simplify the fraction

Now, the fraction becomes \(\frac{(x - 4)(x + 3)}{(5x - 1)(x + 3)}\). Cancel out the common factor \((x + 3)\) (assuming \(x
eq - 3\)).

Answer:

\(\frac{x - 4}{5x - 1}\) (for \(x
eq - 3\))