Answer
$\dfrac{6x - 11}{(x - 2)(x + 2)}$
Restrictions: $x \ne -2, 2$
Work Step by Step
Before adding the two expressions, we need to find the least common denominator. Let's factor the denominators completely first:
$\dfrac{1}{(x - 2)(x + 2)} + \frac{6}{x + 2}$
The least common denominator (LCD) is $(x - 2)(x + 2)$. We have to multiply the numerators by the factor in the LCD that is missing:
$\dfrac{1}{(x - 2)(x + 2)} + \frac{6(x - 2)}{x + 2}$
Multiply out the terms to simplify:
$\dfrac{1}{(x - 2)(x + 2)} + \frac{6x - 12}{(x - 2)(x + 2)}$
Now, we add the numerators and keep the denominator as-is:
$\dfrac{6x - 11}{(x - 2)(x + 2)}$
Now, we need to find the restrictions by seeing what values of the variable will make the denominator equal to zero because if the denominator of any rational expression is zero, the expression is undefined.
To find the restrictions on the variable, set each factor in the denominator equal to zero:
First factor:
$x - 2 = 0$
Add $2$ to each side:
$x = 2$
Second factor:
$x - 2 = 0$
Subtract $2$ from each side:
$x = -2$
Restrictions: $x \ne -2, 2$
