Simplify x+2/x-2 + x-2/x+2 - 3x^2-3/x^2+4

Question

Simplify \(\cfrac{x+2}{x-2}\) + \(\cfrac{x-2}{x+2}\) - \(\cfrac{3x^2-3}{x^2+4}\)

A) \(-\cfrac{x^4+21x^2+20}{x^4-16}\)

B) \(\cfrac{x^4+31x^2+20}{x^4-16}\)

C) \(\cfrac{x^4+31x^2-20}{x^4-16}\)

D) \(\cfrac{-x^4+31x^2+20}{x^4-16}\)

Answer 1

Correct option is (D) \(\frac{-x^4 + 31x^2 +20}{x^4 -16}\)

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

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

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

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

\(=\frac{2(x^4+8x^2+16)-3(x^4-5x^2+4)}{x^4-16}\)

\(=\frac{-x^4 + 31x^2 +20}{x^4 -16}\)

Answer 2

Correct option is D) \(\cfrac{-x^4+31x^2+20}{x^4-16}\)