Question

simplify.
\\(\frac{5w^2 - 20}{w^2 + 9w + 14}\\)

Explanation:

Step1: Factor numerator and denominator

First, factor the numerator \(5w^2 - 20\). We can factor out a 5 first: \(5(w^2 - 4)\). Then, \(w^2 - 4\) is a difference of squares, so it factors to \((w - 2)(w + 2)\). So the numerator becomes \(5(w - 2)(w + 2)\).

Next, factor the denominator \(w^2 + 9w + 14\). We need two numbers that multiply to 14 and add to 9. Those numbers are 2 and 7. So the denominator factors to \((w + 2)(w + 7)\).

Now the expression is \(\frac{5(w - 2)(w + 2)}{(w + 2)(w + 7)}\).

Step2: Cancel common factors

We can cancel out the common factor of \((w + 2)\) from the numerator and the denominator (assuming \(w
eq -2\) to avoid division by zero).

After canceling, we are left with \(\frac{5(w - 2)}{w + 7}\), which can be expanded as \(\frac{5w - 10}{w + 7}\).

Answer:

\(\frac{5(w - 2)}{w + 7}\) (or \(\frac{5w - 10}{w + 7}\))