Simplify 3/(y+3)+(2y)/(y^2+8x+15)

$\frac{3}{y + 3} + \frac{2 y}{y^{2} + 8 x + 15}$

Step 1

To write $\frac{3}{y + 3}$ as a fraction with a common denominator, multiply by $\frac{y^{2} + 8 x + 15}{y^{2} + 8 x + 15}$ .

$\frac{3}{y + 3} \cdot \frac{y^{2} + 8 x + 15}{y^{2} + 8 x + 15} + \frac{2 y}{y^{2} + 8 x + 15}$

Step 2

To write $\frac{2 y}{y^{2} + 8 x + 15}$ as a fraction with a common denominator, multiply by $\frac{y + 3}{y + 3}$ .

$\frac{3}{y + 3} \cdot \frac{y^{2} + 8 x + 15}{y^{2} + 8 x + 15} + \frac{2 y}{y^{2} + 8 x + 15} \cdot \frac{y + 3}{y + 3}$

Step 3

Write each expression with a common denominator of $(y + 3) (y^{2} + 8 x + 15)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{y + 3}$ by $\frac{y^{2} + 8 x + 15}{y^{2} + 8 x + 15}$ .

$\frac{3 (y^{2} + 8 x + 15)}{(y + 3) (y^{2} + 8 x + 15)} + \frac{2 y}{y^{2} + 8 x + 15} \cdot \frac{y + 3}{y + 3}$

Multiply $\frac{2 y}{y^{2} + 8 x + 15}$ by $\frac{y + 3}{y + 3}$ .

$\frac{3 (y^{2} + 8 x + 15)}{(y + 3) (y^{2} + 8 x + 15)} + \frac{2 y (y + 3)}{(y^{2} + 8 x + 15) (y + 3)}$

Reorder the factors of $(y^{2} + 8 x + 15) (y + 3)$ .

$\frac{3 (y^{2} + 8 x + 15)}{(y + 3) (y^{2} + 8 x + 15)} + \frac{2 y (y + 3)}{(y + 3) (y^{2} + 8 x + 15)}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 (y^{2} + 8 x + 15) + 2 y (y + 3)}{(y + 3) (y^{2} + 8 x + 15)}$

Step 5

Simplify the numerator.

Apply the distributive property.

$\frac{3 y^{2} + 3 (8 x) + 3 \cdot 15 + 2 y (y + 3)}{(y + 3) (y^{2} + 8 x + 15)}$

Simplify.

Multiply $8$ by $3$ .

$\frac{3 y^{2} + 24 x + 3 \cdot 15 + 2 y (y + 3)}{(y + 3) (y^{2} + 8 x + 15)}$

Multiply $3$ by $15$ .

$\frac{3 y^{2} + 24 x + 45 + 2 y (y + 3)}{(y + 3) (y^{2} + 8 x + 15)}$

Apply the distributive property.

$\frac{3 y^{2} + 24 x + 45 + 2 y \cdot y + 2 y \cdot 3}{(y + 3) (y^{2} + 8 x + 15)}$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$\frac{3 y^{2} + 24 x + 45 + 2 (y \cdot y) + 2 y \cdot 3}{(y + 3) (y^{2} + 8 x + 15)}$

Multiply $y$ by $y$ .

$\frac{3 y^{2} + 24 x + 45 + 2 y^{2} + 2 y \cdot 3}{(y + 3) (y^{2} + 8 x + 15)}$

Multiply $3$ by $2$ .

$\frac{3 y^{2} + 24 x + 45 + 2 y^{2} + 6 y}{(y + 3) (y^{2} + 8 x + 15)}$

Add $3 y^{2}$ and $2 y^{2}$ .

$\frac{5 y^{2} + 24 x + 45 + 6 y}{(y + 3) (y^{2} + 8 x + 15)}$