Simplify (-2y)/(2y^2+5y)+(y-3)/(2y^2+y-10)

$\frac{- 2 y}{2 y^{2} + 5 y} + \frac{y - 3}{2 y^{2} + y - 10}$

Simplify each term.

Factor $y$ out of $2 y^{2} + 5 y$ .

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

$\frac{- 2 y}{y (2 y) + 5 y} + \frac{y - 3}{2 y^{2} + y - 10}$

Factor $y$ out of $5 y$ .

$\frac{- 2 y}{y (2 y) + y \cdot 5} + \frac{y - 3}{2 y^{2} + y - 10}$

Factor $y$ out of $y (2 y) + y \cdot 5$ .

$\frac{- 2 y}{y (2 y + 5)} + \frac{y - 3}{2 y^{2} + y - 10}$

Cancel the common factor of $y$ .

Cancel the common factor.

$\frac{- 2 y}{y (2 y + 5)} + \frac{y - 3}{2 y^{2} + y - 10}$

Rewrite the expression.

$\frac{- 2}{2 y + 5} + \frac{y - 3}{2 y^{2} + y - 10}$

Move the negative in front of the fraction.

$- \frac{2}{2 y + 5} + \frac{y - 3}{2 y^{2} + y - 10}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 10 = - 20$ and whose sum is $b = 1$ .

Multiply by $1$ .

$- \frac{2}{2 y + 5} + \frac{y - 3}{2 y^{2} + 1 y - 10}$

Rewrite $1$ as $- 4$ plus $5$

$- \frac{2}{2 y + 5} + \frac{y - 3}{2 y^{2} + (- 4 + 5) y - 10}$

Apply the distributive property.

$- \frac{2}{2 y + 5} + \frac{y - 3}{2 y^{2} - 4 y + 5 y - 10}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- \frac{2}{2 y + 5} + \frac{y - 3}{(2 y^{2} - 4 y) + 5 y - 10}$

Factor out the greatest common factor (GCF) from each group.

$- \frac{2}{2 y + 5} + \frac{y - 3}{2 y (y - 2) + 5 (y - 2)}$

Factor the polynomial by factoring out the greatest common factor, $y - 2$ .

$- \frac{2}{2 y + 5} + \frac{y - 3}{(y - 2) (2 y + 5)}$

To write $- \frac{2}{2 y + 5}$ as a fraction with a common denominator, multiply by $\frac{y - 2}{y - 2}$ .

$- \frac{2}{2 y + 5} \cdot \frac{y - 2}{y - 2} + \frac{y - 3}{(y - 2) (2 y + 5)}$

Write each expression with a common denominator of $(2 y + 5) (y - 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{2 y + 5}$ by $\frac{y - 2}{y - 2}$ .

$- \frac{2 (y - 2)}{(2 y + 5) (y - 2)} + \frac{y - 3}{(y - 2) (2 y + 5)}$

Reorder the factors of $(y - 2) (2 y + 5)$ .

$- \frac{2 (y - 2)}{(2 y + 5) (y - 2)} + \frac{y - 3}{(2 y + 5) (y - 2)}$

Combine the numerators over the common denominator.

$\frac{- 2 (y - 2) + y - 3}{(2 y + 5) (y - 2)}$

Simplify the numerator.

Apply the distributive property.

$\frac{- 2 y - 2 \cdot - 2 + y - 3}{(2 y + 5) (y - 2)}$

Multiply $- 2$ by $- 2$ .

$\frac{- 2 y + 4 + y - 3}{(2 y + 5) (y - 2)}$

Add $- 2 y$ and $y$ .

$\frac{- y + 4 - 3}{(2 y + 5) (y - 2)}$

Subtract $3$ from $4$ .

$\frac{- y + 1}{(2 y + 5) (y - 2)}$

Simplify with factoring out.

Factor $- 1$ out of $- y$ .

$\frac{- (y) + 1}{(2 y + 5) (y - 2)}$

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (y) - 1 (- 1)}{(2 y + 5) (y - 2)}$

Factor $- 1$ out of $- (y) - 1 (- 1)$ .

$\frac{- (y - 1)}{(2 y + 5) (y - 2)}$

Simplify the expression.

Rewrite $- (y - 1)$ as $- 1 (y - 1)$ .

$\frac{- 1 (y - 1)}{(2 y + 5) (y - 2)}$

Move the negative in front of the fraction.

$- \frac{y - 1}{(2 y + 5) (y - 2)}$

Simplify (-2y)/(2y^2+5y)+(y-3)/(2y^2+y-10)

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