Simplify 7/(2w^2-19w+9)-(4w+3)/(2w^2+w-1)

$\frac{7}{2 w^{2} - 19 w + 9} - \frac{4 w + 3}{2 w^{2} + w - 1}$

Simplify each term.

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 $a \cdot c = 2 \cdot 9 = 18$ and whose sum is $b = - 19$ .

Factor $- 19$ out of $- 19 w$ .

$\frac{7}{2 w^{2} - 19 (w) + 9} - \frac{4 w + 3}{2 w^{2} + w - 1}$

Rewrite $- 19$ as $- 1$ plus $- 18$

$\frac{7}{2 w^{2} + (- 1 - 18) w + 9} - \frac{4 w + 3}{2 w^{2} + w - 1}$

Apply the distributive property.

$\frac{7}{2 w^{2} - 1 w - 18 w + 9} - \frac{4 w + 3}{2 w^{2} + w - 1}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{7}{(2 w^{2} - 1 w) - 18 w + 9} - \frac{4 w + 3}{2 w^{2} + w - 1}$

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

$\frac{7}{w (2 w - 1) - 9 (2 w - 1)} - \frac{4 w + 3}{2 w^{2} + w - 1}$

Factor the polynomial by factoring out the greatest common factor, $2 w - 1$ .

$\frac{7}{(2 w - 1) (w - 9)} - \frac{4 w + 3}{2 w^{2} + w - 1}$

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 $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = 1$ .

Multiply by $1$ .

$\frac{7}{(2 w - 1) (w - 9)} - \frac{4 w + 3}{2 w^{2} + 1 w - 1}$

Rewrite $1$ as $- 1$ plus $2$

$\frac{7}{(2 w - 1) (w - 9)} - \frac{4 w + 3}{2 w^{2} + (- 1 + 2) w - 1}$

Apply the distributive property.

$\frac{7}{(2 w - 1) (w - 9)} - \frac{4 w + 3}{2 w^{2} - 1 w + 2 w - 1}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{7}{(2 w - 1) (w - 9)} - \frac{4 w + 3}{(2 w^{2} - 1 w) + 2 w - 1}$

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

$\frac{7}{(2 w - 1) (w - 9)} - \frac{4 w + 3}{w (2 w - 1) + 1 (2 w - 1)}$

Factor the polynomial by factoring out the greatest common factor, $2 w - 1$ .

$\frac{7}{(2 w - 1) (w - 9)} - \frac{4 w + 3}{(2 w - 1) (w + 1)}$

To write $\frac{7}{(2 w - 1) (w - 9)}$ as a fraction with a common denominator, multiply by $\frac{w + 1}{w + 1}$ .

$\frac{7}{(2 w - 1) (w - 9)} \cdot \frac{w + 1}{w + 1} - \frac{4 w + 3}{(2 w - 1) (w + 1)}$

To write $- \frac{4 w + 3}{(2 w - 1) (w + 1)}$ as a fraction with a common denominator, multiply by $\frac{w - 9}{w - 9}$ .

$\frac{7}{(2 w - 1) (w - 9)} \cdot \frac{w + 1}{w + 1} - \frac{4 w + 3}{(2 w - 1) (w + 1)} \cdot \frac{w - 9}{w - 9}$

Write each expression with a common denominator of $(2 w - 1) (w - 9) (w + 1)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{7}{(2 w - 1) (w - 9)}$ by $\frac{w + 1}{w + 1}$ .

$\frac{7 (w + 1)}{(2 w - 1) (w - 9) (w + 1)} - \frac{4 w + 3}{(2 w - 1) (w + 1)} \cdot \frac{w - 9}{w - 9}$

Multiply $\frac{4 w + 3}{(2 w - 1) (w + 1)}$ by $\frac{w - 9}{w - 9}$ .

$\frac{7 (w + 1)}{(2 w - 1) (w - 9) (w + 1)} - \frac{(4 w + 3) (w - 9)}{(2 w - 1) (w + 1) (w - 9)}$

Reorder the factors of $(2 w - 1) (w - 9) (w + 1)$ .

$\frac{7 (w + 1)}{(2 w - 1) (w + 1) (w - 9)} - \frac{(4 w + 3) (w - 9)}{(2 w - 1) (w + 1) (w - 9)}$

Combine the numerators over the common denominator.

$\frac{7 (w + 1) - (4 w + 3) (w - 9)}{(2 w - 1) (w + 1) (w - 9)}$

Simplify the numerator.

Apply the distributive property.

$\frac{7 w + 7 \cdot 1 - (4 w + 3) (w - 9)}{(2 w - 1) (w + 1) (w - 9)}$

Multiply $7$ by $1$ .

$\frac{7 w + 7 - (4 w + 3) (w - 9)}{(2 w - 1) (w + 1) (w - 9)}$

Apply the distributive property.

$\frac{7 w + 7 + (- (4 w) - 1 \cdot 3) (w - 9)}{(2 w - 1) (w + 1) (w - 9)}$

Multiply $4$ by $- 1$ .

$\frac{7 w + 7 + (- 4 w - 1 \cdot 3) (w - 9)}{(2 w - 1) (w + 1) (w - 9)}$

Multiply $- 1$ by $3$ .

$\frac{7 w + 7 + (- 4 w - 3) (w - 9)}{(2 w - 1) (w + 1) (w - 9)}$

Expand $(- 4 w - 3) (w - 9)$ using the FOIL Method.

Apply the distributive property.

$\frac{7 w + 7 - 4 w (w - 9) - 3 (w - 9)}{(2 w - 1) (w + 1) (w - 9)}$

Apply the distributive property.

$\frac{7 w + 7 - 4 w \cdot w - 4 w \cdot - 9 - 3 (w - 9)}{(2 w - 1) (w + 1) (w - 9)}$

Apply the distributive property.

$\frac{7 w + 7 - 4 w \cdot w - 4 w \cdot - 9 - 3 w - 3 \cdot - 9}{(2 w - 1) (w + 1) (w - 9)}$

Simplify and combine like terms.

Simplify each term.

Multiply $w$ by $w$ by adding the exponents.

Move $w$ .

$\frac{7 w + 7 - 4 (w \cdot w) - 4 w \cdot - 9 - 3 w - 3 \cdot - 9}{(2 w - 1) (w + 1) (w - 9)}$

Multiply $w$ by $w$ .

$\frac{7 w + 7 - 4 w^{2} - 4 w \cdot - 9 - 3 w - 3 \cdot - 9}{(2 w - 1) (w + 1) (w - 9)}$

Multiply $- 9$ by $- 4$ .

$\frac{7 w + 7 - 4 w^{2} + 36 w - 3 w - 3 \cdot - 9}{(2 w - 1) (w + 1) (w - 9)}$

Multiply $- 3$ by $- 9$ .

$\frac{7 w + 7 - 4 w^{2} + 36 w - 3 w + 27}{(2 w - 1) (w + 1) (w - 9)}$

Subtract $3 w$ from $36 w$ .

$\frac{7 w + 7 - 4 w^{2} + 33 w + 27}{(2 w - 1) (w + 1) (w - 9)}$

Add $7 w$ and $33 w$ .

$\frac{40 w + 7 - 4 w^{2} + 27}{(2 w - 1) (w + 1) (w - 9)}$

Add $7$ and $27$ .

$\frac{40 w - 4 w^{2} + 34}{(2 w - 1) (w + 1) (w - 9)}$

Reorder terms.

$\frac{- 4 w^{2} + 40 w + 34}{(2 w - 1) (w + 1) (w - 9)}$

Factor $2$ out of $- 4 w^{2} + 40 w + 34$ .

Factor $2$ out of $- 4 w^{2}$ .

$\frac{2 (- 2 w^{2}) + 40 w + 34}{(2 w - 1) (w + 1) (w - 9)}$

Factor $2$ out of $40 w$ .

$\frac{2 (- 2 w^{2}) + 2 (20 w) + 34}{(2 w - 1) (w + 1) (w - 9)}$

Factor $2$ out of $34$ .

$\frac{2 (- 2 w^{2}) + 2 (20 w) + 2 (17)}{(2 w - 1) (w + 1) (w - 9)}$

Factor $2$ out of $2 (- 2 w^{2}) + 2 (20 w)$ .

$\frac{2 (- 2 w^{2} + 20 w) + 2 (17)}{(2 w - 1) (w + 1) (w - 9)}$

Factor $2$ out of $2 (- 2 w^{2} + 20 w) + 2 (17)$ .

$\frac{2 (- 2 w^{2} + 20 w + 17)}{(2 w - 1) (w + 1) (w - 9)}$

Simplify with factoring out.

Factor $- 1$ out of $- 2 w^{2}$ .

$\frac{2 (- (2 w^{2}) + 20 w + 17)}{(2 w - 1) (w + 1) (w - 9)}$

Factor $- 1$ out of $20 w$ .

$\frac{2 (- (2 w^{2}) - (- 20 w) + 17)}{(2 w - 1) (w + 1) (w - 9)}$

Factor $- 1$ out of $- (2 w^{2}) - (- 20 w)$ .

$\frac{2 (- (2 w^{2} - 20 w) + 17)}{(2 w - 1) (w + 1) (w - 9)}$

Rewrite $17$ as $- 1 (- 17)$ .

$\frac{2 (- (2 w^{2} - 20 w) - 1 (- 17))}{(2 w - 1) (w + 1) (w - 9)}$

Factor $- 1$ out of $- (2 w^{2} - 20 w) - 1 (- 17)$ .

$\frac{2 (- (2 w^{2} - 20 w - 17))}{(2 w - 1) (w + 1) (w - 9)}$

Simplify the expression.

Rewrite $- (2 w^{2} - 20 w - 17)$ as $- 1 (2 w^{2} - 20 w - 17)$ .

$\frac{2 (- 1 (2 w^{2} - 20 w - 17))}{(2 w - 1) (w + 1) (w - 9)}$

Move the negative in front of the fraction.

$- \frac{2 (2 w^{2} - 20 w - 17)}{(2 w - 1) (w + 1) (w - 9)}$

Simplify 7/(2w^2-19w+9)-(4w+3)/(2w^2+w-1)

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