Simplify (12y^2+12y-7)/(-4y+3)-(y+8)/(4y-3)

$\frac{12 y^{2} + 12 y - 7}{- 4 y + 3} - \frac{y + 8}{4 y - 3}$

Step 1

Factor $- 1$ out of $4 y$ .

$\frac{12 y^{2} + 12 y - 7}{- 4 y + 3} - \frac{y + 8}{- (- 4 y) - 3}$

Step 2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{12 y^{2} + 12 y - 7}{- 4 y + 3} - \frac{y + 8}{- (- 4 y) - 1 (3)}$

Step 3

Factor $- 1$ out of $- (- 4 y) - 1 (3)$ .

$\frac{12 y^{2} + 12 y - 7}{- 4 y + 3} - \frac{y + 8}{- (- 4 y + 3)}$

Step 4

To write $\frac{12 y^{2} + 12 y - 7}{- 4 y + 3}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{12 y^{2} + 12 y - 7}{- 4 y + 3} \cdot \frac{- 1}{- 1} - \frac{y + 8}{- (- 4 y + 3)}$

Step 5

Write each expression with a common denominator of $(- 4 y + 3) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{12 y^{2} + 12 y - 7}{- 4 y + 3}$ by $\frac{- 1}{- 1}$ .

$\frac{(12 y^{2} + 12 y - 7) \cdot - 1}{(- 4 y + 3) \cdot - 1} - \frac{y + 8}{- (- 4 y + 3)}$

Reorder the factors of $(- 4 y + 3) \cdot - 1$ .

$\frac{(12 y^{2} + 12 y - 7) \cdot - 1}{- (- 4 y + 3)} - \frac{y + 8}{- (- 4 y + 3)}$

Step 6

Combine the numerators over the common denominator.

$\frac{(12 y^{2} + 12 y - 7) \cdot - 1 - (y + 8)}{- (- 4 y + 3)}$

Step 7

Simplify the numerator.

Apply the distributive property.

$\frac{12 y^{2} \cdot - 1 + 12 y \cdot - 1 - 7 \cdot - 1 - (y + 8)}{- (- 4 y + 3)}$

Simplify.

Multiply $- 1$ by $12$ .

$\frac{- 12 y^{2} + 12 y \cdot - 1 - 7 \cdot - 1 - (y + 8)}{- (- 4 y + 3)}$

Multiply $- 1$ by $12$ .

$\frac{- 12 y^{2} - 12 y - 7 \cdot - 1 - (y + 8)}{- (- 4 y + 3)}$

Multiply $- 7$ by $- 1$ .

$\frac{- 12 y^{2} - 12 y + 7 - (y + 8)}{- (- 4 y + 3)}$

Apply the distributive property.

$\frac{- 12 y^{2} - 12 y + 7 - y - 1 \cdot 8}{- (- 4 y + 3)}$

Multiply $- 1$ by $8$ .

$\frac{- 12 y^{2} - 12 y + 7 - y - 8}{- (- 4 y + 3)}$

Subtract $y$ from $- 12 y$ .

$\frac{- 12 y^{2} - 13 y + 7 - 8}{- (- 4 y + 3)}$

Subtract $8$ from $7$ .

$\frac{- 12 y^{2} - 13 y - 1}{- (- 4 y + 3)}$

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 $a \cdot c = - 12 \cdot - 1 = 12$ and whose sum is $b = - 13$ .

Factor $- 13$ out of $- 13 y$ .

$\frac{- 12 y^{2} - 13 (y) - 1}{- (- 4 y + 3)}$

Rewrite $- 13$ as $- 1$ plus $- 12$

$\frac{- 12 y^{2} + (- 1 - 12) y - 1}{- (- 4 y + 3)}$

Apply the distributive property.

$\frac{- 12 y^{2} - 1 y - 12 y - 1}{- (- 4 y + 3)}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- 12 y^{2} - 1 y) - 12 y - 1}{- (- 4 y + 3)}$

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

$\frac{y (- 12 y - 1) + 1 (- 12 y - 1)}{- (- 4 y + 3)}$

Factor the polynomial by factoring out the greatest common factor, $- 12 y - 1$ .

$\frac{(- 12 y - 1) (y + 1)}{- (- 4 y + 3)}$

Step 8

Simplify terms.

Move the negative in front of the fraction.

$- \frac{(- 12 y - 1) (y + 1)}{- 4 y + 3}$

Factor $- 1$ out of $- 12 y$ .

$- \frac{(- (12 y) - 1) (y + 1)}{- 4 y + 3}$

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

$- \frac{(- (12 y) - 1 (1)) (y + 1)}{- 4 y + 3}$

Factor $- 1$ out of $- (12 y) - 1 (1)$ .

$- \frac{- (12 y + 1) (y + 1)}{- 4 y + 3}$

Rewrite $- (12 y + 1)$ as $- 1 (12 y + 1)$ .

$- \frac{- 1 (12 y + 1) (y + 1)}{- 4 y + 3}$

Factor $- 1$ out of $- 4 y$ .

$- \frac{- 1 (12 y + 1) (y + 1)}{- (4 y) + 3}$

Rewrite $3$ as $- 1 (- 3)$ .

$- \frac{- 1 (12 y + 1) (y + 1)}{- (4 y) - 1 (- 3)}$

Factor $- 1$ out of $- (4 y) - 1 (- 3)$ .

$- \frac{- 1 (12 y + 1) (y + 1)}{- (4 y - 3)}$

Rewrite $- (4 y - 3)$ as $- 1 (4 y - 3)$ .

$- \frac{- 1 (12 y + 1) (y + 1)}{- 1 (4 y - 3)}$

Cancel the common factor.

$- \frac{- 1 (12 y + 1) (y + 1)}{- 1 (4 y - 3)}$

Rewrite the expression.

$- \frac{(12 y + 1) (y + 1)}{4 y - 3}$

Simplify (12y^2+12y-7)/(-4y+3)-(y+8)/(4y-3)

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