Question 16575
Ok, here we go. I hope you're good at factoring binomials and trinomials.

{{{(U^6 + 2U^3V^3 + V^6)/(U^3 + U^2V - UV^2 - V^3)}}}

LET'S DO THE NUMERATOR FIRST: FACTOR IT.

{{{U^6 + 2U^3V^3 + V^6 = (U^3 + V^3)(U^3 + V^3)}}} NOW FACTOR THE FACTORS WHICH ARE BOTH THE SUM OF TWO CUBES.
{{{(U^3 + V^3)(U^3 + V^3) = (U + V)(U^2 - UV + V^2)(U + V)(U^2 - UV + V^2)}}}

NOW WE'LL DO THE DENOMINATOR. FACTOR BY GROUPING.

{{{(U^3 + U^2V) - (UV^2 + V^3) = U^2(U + V) - V^2( U + V)}}} FACTOR THE (U + V)
{{{(U + V)(U^2 - V^2)}}} NOW FACTOR THE (U^2 - V^2), IT'S THE DIFFERENCE OF TWO SQUARES.

{{{(U + V)(U + V)(U - V)}}} OK, NOW WE CAN PUT IT ALL BACK TOGETHER.

{{{((U + V)(U + V)(U^2 - UV + V^2)(U^2 - UV + V^2))/ (U + V)(U + V)(U - V)}}}

NOW, CANCEL THE (U + V)'S LEAVING:

{{{((U^2 - UV + V^2)(U^2 - UV + V^2))/(U - V)}}} = {{{(U^2 - UV + V^2)^2/(U - V)}}} WE'RE DONE!