Question

Simplify the expression: (3/(4x^2+4x+1)) + ((x+1)/(4x^2-1))

Answer 1

The simplified form of the expression is $\frac{2x^2+9x-2}{(2x+1)^2(2x-1)}$

Given the expression below;

$\frac{3}{4x^2+4x+1} +\frac{x+1}{4x^2-1}$

Simplify the quadratic functions to have:

$=\frac{3}{4x^2+2x+2x+1} +\frac{x+1}{(2x)^2-1^2} \\=\frac{3}{2x(x+1)+1(2x+1)} +\frac{x+1}{(2x+1)(2x-1)} \\=\frac{3}{(2x+1)(2x+1)} +\frac{x+1}{(2x+1)(2x-1)} \\=\frac{3(2x-1)+(x+1)(2x+1)}{(2x+1)^2(2x-1)} \\=\frac{6x-3+(2x^2+3x+1)}{(2x+1)^2(2x-1)} \\=\frac{6x-3+2x^2+3x+1}{(2x+1)^2(2x-1)} \\=\frac{2x^2+9x-2}{(2x+1)^2(2x-1)}$

Hence the simplified form of the expression is $\frac{2x^2+9x-2}{(2x+1)^2(2x-1)}$

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