Question

simplify. \\(\frac{2x^2 + 18x + 16}{x^2 + 5x + 4}\\)

Explanation:

Step1: Factor numerator and denominator

First, factor the numerator \(2x^2 + 18x + 16\). We can factor out a 2 first: \(2(x^2 + 9x + 8)\). Then factor the quadratic inside the parentheses: \(x^2 + 9x + 8=(x + 1)(x + 8)\), so the numerator becomes \(2(x + 1)(x + 8)\).

Next, factor the denominator \(x^2 + 5x + 4\). We need two numbers that multiply to 4 and add to 5, which are 1 and 4. So, \(x^2 + 5x + 4=(x + 1)(x + 4)\).

Step2: Cancel common factors

Now, the expression is \(\frac{2(x + 1)(x + 8)}{(x + 1)(x + 4)}\). We can cancel out the common factor \((x + 1)\) (assuming \(x
eq - 1\) to avoid division by zero).

After canceling, we get \(\frac{2(x + 8)}{x + 4}\) or we can expand the numerator: \(\frac{2x + 16}{x + 4}\).

Answer:

\(\frac{2x + 16}{x + 4}\) (or \(\frac{2(x + 8)}{x + 4}\))