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Answer»

\frac{10x + 3}{3 (x + 1)(2x + 3)}

QUESTION :

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

SOLUTION :

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

Let's find the ROOTS of QUADRATIC equation.

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

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

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

Cross Multiplying the terms

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

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

= \frac{6 {x}^{2} + 3x - 3 + 4 {x}^{2} + 4x + 6x + 6}{(2x + 3)(x + 1)(3x + 3)}

= \frac{6 {x}^{2} + 4 {x}^{2} + 3x + 4x + 6x + 6 - 3}{(2x + 3)(x + 1)(3x + 3)}

= \frac{10 {x}^{2} + 13x + 3 }{(2x + 3)(x + 1)(3x + 3)}

Let's find the roots of quadratic equation,

= \frac{10 {x}^{2} + 10x + 3x + 3 }{(2x + 3)(x + 1)(3x + 1)}

= \frac{10x(x + 1) + 3(x + 1)}{(2x + 3)(x + 1)(3x + 1)}

= \frac{(10x + 3)(x + 1)}{(2x + 3)(x + 1)(3x + 1)}

= \frac{(10x + 3)}{(2x + 3)(3x + 3)}

= \frac{10x + 3}{3 (x + 1)(2x + 3)}


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