What is 2/(2x^2+5x+3)-1/(2x^2-x-6)+3/(x^2-x-2) ?

The solution to 2/(2x^2+5x+3)-1/(2x^2-x-6)+3/(x^2-x-2) is (7x+4)/((x+1)(x-2)(2x+3))

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simplify 2/(2x^2+5x+3)-1/(2x^2-x-6)+3/(x^2-x-2)

simplify

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

\frac{7 x + 4}{( x + 1 ) ( x - 2 ) ( 2 x + 3 )}

Solution steps

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

Least Common Multiplier of

2 x^{2} + 5 x + 3, 2 x^{2} - x - 6, x^{2} - x - 2 : (x + 1) (x - 2) (2 x + 3)

Adjust Fractions based on the LCM

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

Apply the fraction rule:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

Simplify

2 (x - 2) - (x + 1) + 3 (2 x + 3) : 7 x + 4

= \frac{7 x + 4}{( x + 1 ) ( x - 2 ) ( 2 x + 3 )}

\frac{x ^{2} - 36}{2 x ^{3} - 12 x ^{2}}

1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}

(x + 9) (x - 1)

(x + 9) (x - 4)

\frac{70}{3}

What is 2/(2x^2+5x+3)-1/(2x^2-x-6)+3/(x^2-x-2) ?
The solution to 2/(2x^2+5x+3)-1/(2x^2-x-6)+3/(x^2-x-2) is (7x+4)/((x+1)(x-2)(2x+3))