Question 171793
{{{(1/(x-x^2)-1/(x^2+x))/(1/(x+1)+1/(x^2-1))}}} Start with the given expression.



{{{(1/(-x(x-1))-1/(x^2+x))/(1/(x+1)+1/(x^2-1))}}} Factor {{{x-x^2}}} to get {{{x(1-x)=-x(x-1)}}}



{{{(-1/(x(x-1))-1/(x^2+x))/(1/(x+1)+1/(x^2-1))}}} Reduce



{{{(-1/(x(x-1))-1/(x(x+1)))/(1/(x+1)+1/(x^2-1))}}} Factor {{{x^2+x}}} to get {{{x(x+1)}}}



{{{(-1/(x(x-1))-1/(x(x+1)))/(1/(x+1)+1/((x-1)(x+1)))}}} Factor {{{x^2-1}}} to get {{{(x-1)(x+1)}}}



Now take note that the LCD of the inner denominators is {{{x(x-1)(x+1)}}}



{{{((-1/(cross(x)cross((x-1))))(cross(x)cross((x-1))(x+1))-(1/(cross(x)cross((x+1))))(cross(x)(x-1)cross((x+1))))/((1/cross((x+1)))(x(x-1)cross((x+1)))+(1/(cross((x-1)(x+1))))(x*cross((x-1)(x+1))))}}} Multiply EVERY term by the inner LCD to clear out the inner fractions.



{{{(-1(x+1)-(x-1))/(x(x-1)+x)}}} Simplify



{{{(-x-1-x+1)/(x^2-x+x)}}} Distribute



{{{(-2x)/(x^2)}}} Combine like terms.



{{{(-2x)/(x*x)}}} Factor {{{x^2}}} to get {{{x*x}}}



{{{(-2*highlight(x))/(highlight(x)*x)}}} Highlight the common terms.



{{{(-2*cross(x))/(cross(x)*x)}}} Cancel out the common terms.



{{{-2/(x)}}} Simplify.



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Answer:



So {{{(1/(x-x^2)-1/(x^2+x))/(1/(x+1)+1/(x^2-1))}}} simplifies to {{{-2/(x)}}}