Answer
$B$
Work Step by Step
Set up the expression as the product of two rational expressions:
$\frac{x^2 + 5x + 4}{(x - 1)(x + 1)} \cdot \frac{x^2 - 5x + 6}{x - 2}$
Factor all expressions completely:
$\frac{(x + 4)(x + 1)}{(x - 1)(x + 1)} \cdot \frac{(x - 3)(x - 2)}{x - 2}$
Multiply to simplify:
$\frac{(x + 4)(x + 1)(x - 3)(x - 2)}{(x - 1)(x + 1)(x - 2)}$
Cancel common factors in the numerator and denominator:
$\frac{(x + 4)(x - 3)}{x - 1}$
Multiply to simplify:
$\frac{x^2 + x - 12}{x - 1}$
Restrictions on $x$ occur when the value of $x$ makes the fraction undefined, which means that the denominator becomes $0$.
Set the factors in the denominators equal to $0$ to find restrictions:
First factor:
$x - 1 = 0$
Subtract $3$ from each side of the equation:
$x = 1$
Second factor:
$x + 1 = 0$
Subtract $1$ from each side of the equation:
$x = -1$
Third factor:
$x - 2 = 0$
Add $2$ to each side of the equation:
$x = 2$
Restriction: $x \ne -1, 1, 2$
This corresponds to answer option $B$.
