Question

simplify.
\\(\frac{y^2 - y - 2}{12 - 3y^2}\\)

Explanation:

Step1: Factor numerator and denominator

Factor \(y^2 - y - 2\): find two numbers that multiply to \(-2\) and add to \(-1\), which are \(-2\) and \(1\). So \(y^2 - y - 2=(y - 2)(y + 1)\).

Factor \(12 - 3y^2\): first factor out \(-3\), we get \(12 - 3y^2=-3(y^2 - 4)\), and \(y^2 - 4\) is a difference of squares, so \(y^2 - 4=(y - 2)(y + 2)\). Thus \(12 - 3y^2=-3(y - 2)(y + 2)\).

Now the fraction becomes \(\frac{(y - 2)(y + 1)}{-3(y - 2)(y + 2)}\).

Step2: Cancel common factors

Cancel out the common factor \((y - 2)\) (assuming \(y
eq2\) to avoid division by zero).

We get \(\frac{y + 1}{-3(y + 2)}=-\frac{y + 1}{3(y + 2)}\) or \(\frac{-y - 1}{3y + 6}\).

Answer:

\(-\dfrac{y + 1}{3(y + 2)}\) (or equivalent forms like \(\dfrac{-y - 1}{3y + 6}\))