Simplify (z^2+4zy+4y^2)/(4z+8y)

$\frac{z^{2} + 4 z y + 4 y^{2}}{4 z + 8 y}$

Factor using the perfect square rule.

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

$\frac{z^{2} + 4 z y + {(2 y)}^{2}}{4 z + 8 y}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 z y = 2 \cdot z \cdot (2 y)$

Rewrite the polynomial.

$\frac{z^{2} + 2 \cdot z \cdot (2 y) + {(2 y)}^{2}}{4 z + 8 y}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = z$ and $b = 2 y$ .

$\frac{{(z + 2 y)}^{2}}{4 z + 8 y}$

Simplify terms.

Factor $4$ out of $4 z + 8 y$ .

Factor $4$ out of $4 z$ .

$\frac{{(z + 2 y)}^{2}}{4 (z) + 8 y}$

Factor $4$ out of $8 y$ .

$\frac{{(z + 2 y)}^{2}}{4 (z) + 4 (2 y)}$

Factor $4$ out of $4 (z) + 4 (2 y)$ .

$\frac{{(z + 2 y)}^{2}}{4 (z + 2 y)}$

Cancel the common factor of ${(z + 2 y)}^{2}$ and $z + 2 y$ .

Factor $z + 2 y$ out of ${(z + 2 y)}^{2}$ .

$\frac{(z + 2 y) (z + 2 y)}{4 (z + 2 y)}$

Cancel the common factors.

Factor $z + 2 y$ out of $4 (z + 2 y)$ .

$\frac{(z + 2 y) (z + 2 y)}{(z + 2 y) \cdot 4}$

Cancel the common factor.

$\frac{(z + 2 y) (z + 2 y)}{(z + 2 y) \cdot 4}$

Rewrite the expression.

$\frac{z + 2 y}{4}$