Expand.
\[\frac{{a}^{2}+2ab+{b}^{2}-3ab}{2{(a-b)}^{2}+4ab}\times \frac{{(a+b)}^{2}+{(a-b)}^{2}}{{a}^{3}+{b}^{3}}\]
Collect like terms.
\[\frac{{a}^{2}+(2ab-3ab)+{b}^{2}}{2{(a-b)}^{2}+4ab}\times \frac{{(a+b)}^{2}+{(a-b)}^{2}}{{a}^{3}+{b}^{3}}\]
Simplify \({a}^{2}+(2ab-3ab)+{b}^{2}\) to \({a}^{2}-ab+{b}^{2}\).
\[\frac{{a}^{2}-ab+{b}^{2}}{2{(a-b)}^{2}+4ab}\times \frac{{(a+b)}^{2}+{(a-b)}^{2}}{{a}^{3}+{b}^{3}}\]
Factor out the common term \(2\).
\[\frac{{a}^{2}-ab+{b}^{2}}{2({(a-b)}^{2}+2ab)}\times \frac{{(a+b)}^{2}+{(a-b)}^{2}}{{a}^{3}+{b}^{3}}\]
Expand.
\[\frac{{a}^{2}-ab+{b}^{2}}{2({a}^{2}-2ab+{b}^{2}+2ab)}\times \frac{{(a+b)}^{2}+{(a-b)}^{2}}{{a}^{3}+{b}^{3}}\]
Collect like terms.
\[\frac{{a}^{2}-ab+{b}^{2}}{2({a}^{2}+(-2ab+2ab)+{b}^{2})}\times \frac{{(a+b)}^{2}+{(a-b)}^{2}}{{a}^{3}+{b}^{3}}\]
Simplify \({a}^{2}+(-2ab+2ab)+{b}^{2}\) to \({a}^{2}+{b}^{2}\).
\[\frac{{a}^{2}-ab+{b}^{2}}{2({a}^{2}+{b}^{2})}\times \frac{{(a+b)}^{2}+{(a-b)}^{2}}{{a}^{3}+{b}^{3}}\]
Expand.
\[\frac{{a}^{2}-ab+{b}^{2}}{2({a}^{2}+{b}^{2})}\times \frac{{a}^{2}+2ab+{b}^{2}+{a}^{2}-2ab+{b}^{2}}{{a}^{3}+{b}^{3}}\]
Collect like terms.
\[\frac{{a}^{2}-ab+{b}^{2}}{2({a}^{2}+{b}^{2})}\times \frac{({a}^{2}+{a}^{2})+(2ab-2ab)+({b}^{2}+{b}^{2})}{{a}^{3}+{b}^{3}}\]
Simplify \(({a}^{2}+{a}^{2})+(2ab-2ab)+({b}^{2}+{b}^{2})\) to \(2{a}^{2}+2{b}^{2}\).
\[\frac{{a}^{2}-ab+{b}^{2}}{2({a}^{2}+{b}^{2})}\times \frac{2{a}^{2}+2{b}^{2}}{{a}^{3}+{b}^{3}}\]
Factor out the common term \(2\).
\[\frac{{a}^{2}-ab+{b}^{2}}{2({a}^{2}+{b}^{2})}\times \frac{2({a}^{2}+{b}^{2})}{{a}^{3}+{b}^{3}}\]
Use Sum of Cubes: \({a}^{3}+{b}^{3}=(a+b)({a}^{2}-ab+{b}^{2})\).
\[\frac{{a}^{2}-ab+{b}^{2}}{2({a}^{2}+{b}^{2})}\times \frac{2({a}^{2}+{b}^{2})}{(a+b)({a}^{2}-(a)(b)+{b}^{2})}\]
Remove parentheses.
\[\frac{{a}^{2}-ab+{b}^{2}}{2({a}^{2}+{b}^{2})}\times \frac{2({a}^{2}+{b}^{2})}{(a+b)({a}^{2}-ab+{b}^{2})}\]
Cancel \({a}^{2}+{b}^{2}\).
\[\frac{{a}^{2}-ab+{b}^{2}}{2}\times \frac{2}{(a+b)({a}^{2}-ab+{b}^{2})}\]
Cancel \(2\).
\[({a}^{2}-ab+{b}^{2})\times \frac{1}{(a+b)({a}^{2}-ab+{b}^{2})}\]
Cancel \({a}^{2}-ab+{b}^{2}\).
\[\frac{1}{a+b}\]
1/(a+b)
