Simplify (y^2+3y)/(y^2+6y+9)

$\frac{y^{2} + 3 y}{y^{2} + 6 y + 9}$

Step 1

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

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

$\frac{y \cdot y + 3 y}{y^{2} + 6 y + 9}$

Factor $y$ out of $3 y$ .

$\frac{y \cdot y + y \cdot 3}{y^{2} + 6 y + 9}$

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

$\frac{y (y + 3)}{y^{2} + 6 y + 9}$

Step 2

Factor using the perfect square rule.

Rewrite $9$ as $3^{2}$ .

$\frac{y (y + 3)}{y^{2} + 6 y + 3^{2}}$

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

$6 y = 2 \cdot y \cdot 3$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 3$ .

$\frac{y (y + 3)}{{(y + 3)}^{2}}$

Step 3

Cancel the common factor of $y + 3$ and ${(y + 3)}^{2}$ .

Factor $y + 3$ out of $y (y + 3)$ .

$\frac{(y + 3) y}{{(y + 3)}^{2}}$

Cancel the common factors.

Factor $y + 3$ out of ${(y + 3)}^{2}$ .

$\frac{(y + 3) y}{(y + 3) (y + 3)}$

Cancel the common factor.

$\frac{(y + 3) y}{(y + 3) (y + 3)}$

Rewrite the expression.

$\frac{y}{y + 3}$