Simplify (z^2-4)/(z^2-1)*(z+1)/(z+2)

$\frac{z^{2} - 4}{z^{2} - 1} \cdot \frac{z + 1}{z + 2}$

Step 1

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

$\frac{z^{2} - 2^{2}}{z^{2} - 1} \cdot \frac{z + 1}{z + 2}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = 2$ .

$\frac{(z + 2) (z - 2)}{z^{2} - 1} \cdot \frac{z + 1}{z + 2}$

Step 2

Simplify the denominator.

Rewrite $1$ as $1^{2}$ .

$\frac{(z + 2) (z - 2)}{z^{2} - 1^{2}} \cdot \frac{z + 1}{z + 2}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = 1$ .

$\frac{(z + 2) (z - 2)}{(z + 1) (z - 1)} \cdot \frac{z + 1}{z + 2}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $z + 2$ .

Cancel the common factor.

$\frac{(z + 2) (z - 2)}{(z + 1) (z - 1)} \cdot \frac{z + 1}{z + 2}$

Rewrite the expression.

$\frac{z - 2}{(z + 1) (z - 1)} \cdot (z + 1)$

Cancel the common factor of $z + 1$ .

Cancel the common factor.

$\frac{z - 2}{(z + 1) (z - 1)} \cdot (z + 1)$

Rewrite the expression.

$\frac{z - 2}{z - 1}$