Question

Simplify the expression given below.x+2/4x²+5x+1*4x+1/x²-4

Answer 1

Answer:  $\bold{\dfrac{1}{(x+1)(x-2)}}$

Step-by-step explanation:

$\dfrac{x+2}{4x^2+5x+1}\times \dfrac{4x+1}{x^2-4}\\\\\\\text{Factor the quadratics:}\\\dfrac{x+2}{(4x+1)(x+1)}\times \dfrac{4x+1}{(x-2)(x+2)}\\\\\\\text{Simplify - cross out (4x+1) and (x+2):}\\\dfrac{1}{(x+1)(x-2)}$

Answer 2

Answer:

$\frac{1}{x^2 - x - 2}$

Step-by-step explanation:

The given expression is

$\frac{x+2}{4x^2+5x+1}\times \frac{4x+1}{x^2-4}$

Factorize the denominators.

$\frac{x+2}{4x^2+4x+x+1}\times \frac{4x+1}{x^2-2^2}$

$\frac{x+2}{4x(x+1)+1(x+1)}\times \frac{4x+1}{(x-2)(x+2)}$             $[\because a^2-b^2=(a-b)(a+b)]$

$\frac{x+2}{(x+1)(4x+1)}\times \frac{4x+1}{(x-2)(x+2)}$

Cancel out common factors.

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

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

On further simplification we get

$\frac{1}{x^2 - x - 2}$

Therefore, the simplified form of the given expression is $\frac{1}{x^2 - x - 2}$.