Simplify (z^2)/(z+1)*(z^2+3z+2)/(z^2+3z)

$\frac{z^{2}}{z + 1} \cdot \frac{z^{2} + 3 z + 2}{z^{2} + 3 z}$

Factor $z^{2} + 3 z + 2$ using the AC method.

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Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Write the factored form using these integers.

$\frac{z^{2}}{z + 1} \cdot \frac{(z + 1) (z + 2)}{z^{2} + 3 z}$

Simplify terms.

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Factor $z$ out of $z^{2} + 3 z$ .

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Factor $z$ out of $z^{2}$ .

$\frac{z^{2}}{z + 1} \cdot \frac{(z + 1) (z + 2)}{z \cdot z + 3 z}$

Factor $z$ out of $3 z$ .

$\frac{z^{2}}{z + 1} \cdot \frac{(z + 1) (z + 2)}{z \cdot z + z \cdot 3}$

Factor $z$ out of $z \cdot z + z \cdot 3$ .

$\frac{z^{2}}{z + 1} \cdot \frac{(z + 1) (z + 2)}{z (z + 3)}$

Cancel the common factor of $z$ .

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Factor $z$ out of $z^{2}$ .

$\frac{z \cdot z}{z + 1} \cdot \frac{(z + 1) (z + 2)}{z (z + 3)}$

Cancel the common factor.

$\frac{z \cdot z}{z + 1} \cdot \frac{(z + 1) (z + 2)}{z (z + 3)}$

Rewrite the expression.

$\frac{z}{z + 1} \cdot \frac{(z + 1) (z + 2)}{z + 3}$

Cancel the common factor of $z + 1$ .

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Cancel the common factor.

$\frac{z}{z + 1} \cdot \frac{(z + 1) (z + 2)}{z + 3}$

Rewrite the expression.

$z \cdot \frac{z + 2}{z + 3}$

Combine $z$ and $\frac{z + 2}{z + 3}$ .

$\frac{z (z + 2)}{z + 3}$

Simplify (z^2)/(z+1)*(z^2+3z+2)/(z^2+3z)

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