Question

rational expressions. add. state the sum in simplest form. $\frac{x^{2}}{x + 2}+\frac{4x}{x + 2}+\frac{4}{x + 2}$, $x
eq - 2$. options: $x + 2$, $x^{2}+2x + 2$, $\frac{x^{2}+4x + 4}{x^{3}+8}$, $x + 4$

Explanation:

Step1: Combine the numerators over the common denominator

Since the denominators of the rational - expressions $\frac{x^{2}}{x + 2}+\frac{4x}{x + 2}+\frac{4}{x + 2}$ are the same ($x+2$), we add the numerators: $\frac{x^{2}+4x + 4}{x + 2}$.

Step2: Factor the numerator

We factor the quadratic expression $x^{2}+4x + 4$ using the perfect - square formula $(a + b)^2=a^{2}+2ab + b^{2}$. Here, $a=x$ and $b = 2$, so $x^{2}+4x + 4=(x + 2)^{2}$. Then our expression becomes $\frac{(x + 2)^{2}}{x+2}$.

Step3: Simplify the rational expression

Cancel out the common factor $(x + 2)$ in the numerator and the denominator (since $x
eq - 2$). We get $x + 2$.

Answer:

$x + 2$