Simplify 3/(2x^2-x-3)-2/(x^2-5x-6)

$\frac{3}{2 x^{2} - x - 3} - \frac{2}{x^{2} - 5 x - 6}$

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- x$ .

$\frac{3}{2 x^{2} - (x) - 3} - \frac{2}{x^{2} - 5 x - 6}$

Rewrite $- 1$ as $2$ plus $- 3$

$\frac{3}{2 x^{2} + (2 - 3) x - 3} - \frac{2}{x^{2} - 5 x - 6}$

Apply the distributive property.

$\frac{3}{2 x^{2} + 2 x - 3 x - 3} - \frac{2}{x^{2} - 5 x - 6}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{3}{(2 x^{2} + 2 x) - 3 x - 3} - \frac{2}{x^{2} - 5 x - 6}$

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

$\frac{3}{2 x (x + 1) - 3 (x + 1)} - \frac{2}{x^{2} - 5 x - 6}$

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$\frac{3}{(x + 1) (2 x - 3)} - \frac{2}{x^{2} - 5 x - 6}$

Factor $x^{2} - 5 x - 6$ 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 $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Write the factored form using these integers.

$\frac{3}{(x + 1) (2 x - 3)} - \frac{2}{(x - 6) (x + 1)}$

To write $\frac{3}{(x + 1) (2 x - 3)}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{3}{(x + 1) (2 x - 3)} \cdot \frac{x - 6}{x - 6} - \frac{2}{(x - 6) (x + 1)}$

To write $- \frac{2}{(x - 6) (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{2 x - 3}{2 x - 3}$ .

$\frac{3}{(x + 1) (2 x - 3)} \cdot \frac{x - 6}{x - 6} - \frac{2}{(x - 6) (x + 1)} \cdot \frac{2 x - 3}{2 x - 3}$

Write each expression with a common denominator of $(x + 1) (2 x - 3) (x - 6)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{(x + 1) (2 x - 3)}$ by $\frac{x - 6}{x - 6}$ .

$\frac{3 (x - 6)}{(x + 1) (2 x - 3) (x - 6)} - \frac{2}{(x - 6) (x + 1)} \cdot \frac{2 x - 3}{2 x - 3}$

Multiply $\frac{2}{(x - 6) (x + 1)}$ by $\frac{2 x - 3}{2 x - 3}$ .

$\frac{3 (x - 6)}{(x + 1) (2 x - 3) (x - 6)} - \frac{2 (2 x - 3)}{(x - 6) (x + 1) (2 x - 3)}$

Reorder the factors of $(x + 1) (2 x - 3) (x - 6)$ .

$\frac{3 (x - 6)}{(2 x - 3) (x + 1) (x - 6)} - \frac{2 (2 x - 3)}{(x - 6) (x + 1) (2 x - 3)}$

Reorder the factors of $(x - 6) (x + 1) (2 x - 3)$ .

$\frac{3 (x - 6)}{(2 x - 3) (x + 1) (x - 6)} - \frac{2 (2 x - 3)}{(2 x - 3) (x + 1) (x - 6)}$

Combine the numerators over the common denominator.

$\frac{3 (x - 6) - 2 (2 x - 3)}{(2 x - 3) (x + 1) (x - 6)}$

Simplify the numerator.

Apply the distributive property.

$\frac{3 x + 3 \cdot - 6 - 2 (2 x - 3)}{(2 x - 3) (x + 1) (x - 6)}$

Multiply $3$ by $- 6$ .

$\frac{3 x - 18 - 2 (2 x - 3)}{(2 x - 3) (x + 1) (x - 6)}$

Apply the distributive property.

$\frac{3 x - 18 - 2 (2 x) - 2 \cdot - 3}{(2 x - 3) (x + 1) (x - 6)}$

Multiply $2$ by $- 2$ .

$\frac{3 x - 18 - 4 x - 2 \cdot - 3}{(2 x - 3) (x + 1) (x - 6)}$

Multiply $- 2$ by $- 3$ .

$\frac{3 x - 18 - 4 x + 6}{(2 x - 3) (x + 1) (x - 6)}$

Subtract $4 x$ from $3 x$ .

$\frac{- x - 18 + 6}{(2 x - 3) (x + 1) (x - 6)}$

Add $- 18$ and $6$ .

$\frac{- x - 12}{(2 x - 3) (x + 1) (x - 6)}$

Simplify with factoring out.

Factor $- 1$ out of $- x$ .

$\frac{- (x) - 12}{(2 x - 3) (x + 1) (x - 6)}$

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

$\frac{- (x) - 1 (12)}{(2 x - 3) (x + 1) (x - 6)}$

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

$\frac{- (x + 12)}{(2 x - 3) (x + 1) (x - 6)}$

Simplify the expression.

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

$\frac{- 1 (x + 12)}{(2 x - 3) (x + 1) (x - 6)}$

Move the negative in front of the fraction.

$- \frac{x + 12}{(2 x - 3) (x + 1) (x - 6)}$

Simplify 3/(2x^2-x-3)-2/(x^2-5x-6)

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