Question

simplify the following expression completely.\\(\frac{x^2 + 19x + 90}{x^2 + 12x + 20}\\)

Explanation:

Step1: Factor the numerator

We need to factor \(x^{2}+19x + 90\). We look for two numbers that multiply to \(90\) and add up to \(19\). The numbers are \(9\) and \(10\) since \(9\times10 = 90\) and \(9 + 10=19\). So, \(x^{2}+19x + 90=(x + 9)(x+10)\).

Step2: Factor the denominator

We factor \(x^{2}+12x + 20\). We look for two numbers that multiply to \(20\) and add up to \(12\). The numbers are \(2\) and \(10\) since \(2\times10 = 20\) and \(2+10 = 12\). So, \(x^{2}+12x + 20=(x + 2)(x + 10)\).

Step3: Simplify the fraction

Now we have \(\frac{(x + 9)(x + 10)}{(x + 2)(x + 10)}\). We can cancel out the common factor \((x + 10)\) (assuming \(x
eq - 10\) to avoid division by zero). So the simplified form is \(\frac{x + 9}{x + 2}\).

Answer:

\(\frac{x + 9}{x + 2}\)