Question

Find the sum of the terms.

Answer 1

First we have to simplify the denominator:x² + 3 x + 2 = x² + 2 x + x + 2 = x ( x + 2 ) + ( x + 2 ) = ( x + 2 ) ( x + 1 )x / ( x + 2 ) ( x + 1 )  +  3 / ( x + 1 ) == x + 3 ( x + 2 ) / ( x + 2 ) ( x + 1 ) == x + 3 x + 6 / ( x + 2 ) ( x + 1 ) = = 4 x + 6 / ( x + 2 ) ( x + 1 )Answer: C ) 4 x + 6

Answer 2

Answer: Option (C) is the correct answer.

Step-by-step explanation:

The given equation is as follows.

        $\frac{x}{x^{2}+3x+2} + \frac{3}{x+1}$

Taking L.C.M, the equation will become as follows.

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

Factorize the equation $x^{2}+3x+2$ in the denominator as follows.

         $x^{2}+3x+2$

       =  $x^{2} + 2x + x + 2$  

       = x(x+2) + 1(x + 2)  

       = (x+1)(x+2) ........ (2)

Put the factors in equation (2) in to equation (1), then the equation will become as follows.

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

     = $\frac{x^{2}+x +3x^{2}+9x+6)}{(x+1)(x+2)(x+1)}$

     = $\frac{4x^{2}+10x+6}{(x+1)^{2}(x+2)}$

Now, factorize the numerator as follows.

       $\frac{4x^{2}+10x+6}{(x+1)^{2}(x+2)}$

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

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

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

Cancelling (x+1) from both numerator and denominator. Then the equation will be written as follows.

       $\frac{(4x+6)(x+1)}{(x+1)^{2}(x+2)}$

     = $\frac{(4x+6)}{(x+1)(x+2)}$

Thus, we can conclude that the numerator of simplified sum is (4x+6).