Question 454769
{{{ 1/(a-b) + 4/(b-a) - 8/(a+b) - (11a-5b) / (b^2-a^2) }}}
First factor the denominator of the last term
{{{ 1/(a-b) + 4/(b-a) - 8/(a+b) - (11a-5b) / ((b - a)*( b + a)) }}}
Now express the first term as: {{{ 1/(a - b) = -1/(b - a) }}}
{{{ -1/(b - a) + 4/(b-a) - 8/(b + a) - (11a-5b) / ((b - a)*( b + a)) }}}
Combine the first two terms:
{{{  -1/(b - a) + 4/(b-a) = (4 - 1) / (b - a) }}}
{{{ (4 - 1) / (b - a) = 3 / (b-a) }}}
So far I have:
{{{ 3 / (b-a) - 8/(b + a) - (11a-5b) / ((b - a)*( b + a)) }}}
Now multiply top and bottom of 1st term  by {{{ (b + a) / (b + a) }}}
Multiply top and bottom of 2nd term by {{{ (b - a)*(b - a) }}}
{{{ (3*(b + a)) /  ((b - a)*( b + a)) - (8*(b - a)) /  ((b - a)*( b + a)) - (11a-5b) / ((b - a)*( b + a)) }}}
Now combine the fractions with common denominator:
{{{  (3*(b + a) - 8*(b - a) - (11a - 5b)) / ((b - a)*(b + a)) }}}
Combine terms in the numerator
{{{ ( 3b + 3a - 8b + 8a - 11a + 5b) / ((b - a)*(b + a)) }}} 
{{{ ( 3a + 8a - 11a + 3b - 8b + 5b ) / ((b - a)*(b + a)) }}} 
The numerator is {{{ 0 }}}, so
{{{ 1/(a-b) + 4/(b-a) - 8/(a+b) - (11a-5b) / (b^2-a^2) = 0 }}}
One way to check the answer is to give values for {{{a}}} and {{{b}}}
Note that {{{a}}} cannot = {{{b}}}, since the answer would be {{{ 0/0 }}}
I'll say {{{ a = 3}}}, {{{ b = 5 }}}
{{{ 1/(a-b) + 4/(b-a) - 8/(a+b) - (11a-5b) / (b^2-a^2) = 0 }}}
{{{ 1/(3 - 5) + 4/(5 - 3) - 8/(3 + 5) - (11*3 - 5*5) / (5^2 - 3^2) = 0 }}}
{{{ 1/(-2) + 4/ 2 - 8/8 - (33 - 25) / (25 - 9) = 0 }}}
{{{ -1/2 + 4/ 2 - 2/2 - 8 / 16 = 0 }}}
{{{ 1/2 - 1/2 = 0 }}}
{{{ 0 = 0 }}}
OK