Question

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Answer 1

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

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,

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

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

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

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

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

Answer 2

Question :-

$\bf \:Solve : \dfrac{2x - 1}{ {2x}^{2} + 5x + 3 } + \dfrac{2}{3x + 3}$

Answer :-

$\bf \:\dfrac{2x - 1}{ {2x}^{2} + 5x + 3 } + \dfrac{2}{3x + 3}$

$\bf\implies \:\dfrac{2x - 1}{ {2x}^{2} + 2x + 3x + 3 } + \dfrac{2}{3(x + 1)}$

$\bf\implies \:\dfrac{2x - 1}{ 2x(x + 1) + 3(x + 1) } + \dfrac{2}{3(x + 1)}$

$\bf\implies \:\dfrac{2x - 1}{ (2x+ 3)(x + 1) } + \dfrac{2}{3(x + 1)}$

On taking LCM, we get

$\bf\implies \:\dfrac{3(2x - 1)+ 2(2x + 3)}{ 3(2x+ 3)(x + 1) }$

$\bf\implies \:\dfrac{6x - 3 + 4x + 6}{ 3(2x+ 3)(x + 1) }$

$\bf\implies \:\dfrac{10x + 3}{ 3(2x+ 3)(x + 1) }$

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