Simplify 12/(3y^2-10y-8)-3/(y^2-6y+8)

$\frac{12}{3 y^{2} - 10 y - 8} - \frac{3}{y^{2} - 6 y + 8}$

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 $a \cdot c = 3 \cdot - 8 = - 24$ and whose sum is $b = - 10$ .

Factor $- 10$ out of $- 10 y$ .

$\frac{12}{3 y^{2} - 10 (y) - 8} - \frac{3}{y^{2} - 6 y + 8}$

Rewrite $- 10$ as $2$ plus $- 12$

$\frac{12}{3 y^{2} + (2 - 12) y - 8} - \frac{3}{y^{2} - 6 y + 8}$

Apply the distributive property.

$\frac{12}{3 y^{2} + 2 y - 12 y - 8} - \frac{3}{y^{2} - 6 y + 8}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{12}{(3 y^{2} + 2 y) - 12 y - 8} - \frac{3}{y^{2} - 6 y + 8}$

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

$\frac{12}{y (3 y + 2) - 4 (3 y + 2)} - \frac{3}{y^{2} - 6 y + 8}$

Factor the polynomial by factoring out the greatest common factor, $3 y + 2$ .

$\frac{12}{(3 y + 2) (y - 4)} - \frac{3}{y^{2} - 6 y + 8}$

Factor $y^{2} - 6 y + 8$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $8$ and whose sum is $- 6$ .

$- 4, - 2$

Write the factored form using these integers.

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

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

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

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

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

Write each expression with a common denominator of $(3 y + 2) (y - 4) (y - 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{12}{(3 y + 2) (y - 4)}$ by $\frac{y - 2}{y - 2}$ .

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

Multiply $\frac{3}{(y - 4) (y - 2)}$ by $\frac{3 y + 2}{3 y + 2}$ .

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

Reorder the factors of $(3 y + 2) (y - 4) (y - 2)$ .

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

Reorder the factors of $(y - 4) (y - 2) (3 y + 2)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Factor $3$ out of $12 (y - 2) - 3 (3 y + 2)$ .

Factor $3$ out of $12 (y - 2)$ .

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

Factor $3$ out of $- 3 (3 y + 2)$ .

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

Factor $3$ out of $3 (4 (y - 2)) + 3 (- (3 y + 2))$ .

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

Apply the distributive property.

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

Multiply $4$ by $- 2$ .

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

Apply the distributive property.

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

Multiply $3$ by $- 1$ .

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

Multiply $- 1$ by $2$ .

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

Subtract $3 y$ from $4 y$ .

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

Subtract $2$ from $- 8$ .

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

Simplify 12/(3y^2-10y-8)-3/(y^2-6y+8)

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