Factor out the common term \(ab\).
\[\frac{{b}^{2}}{{a}^{2}+ab+{b}^{2}}+\frac{ab(4a-b)}{{b}^{3}-{a}^{3}}+\frac{a}{a-b}\times 3a-2+4a\times 3\]
Use Difference of Cubes : \({a}^{3}-{b}^{3}=(a-b)({a}^{2}+ab+{b}^{2})\).
\[\frac{{b}^{2}}{{a}^{2}+ab+{b}^{2}}+\frac{ab(4a-b)}{(b-a)({b}^{2}+(b)(a)+{a}^{2})}+\frac{a}{a-b}\times 3a-2+4a\times 3\]
Remove parentheses.
\[\frac{{b}^{2}}{{a}^{2}+ab+{b}^{2}}+\frac{ab(4a-b)}{(b-a)({b}^{2}+ba+{a}^{2})}+\frac{a}{a-b}\times 3a-2+4a\times 3\]
Simplify \(\frac{a}{a-b}\times 3a\) to \(\frac{3{a}^{2}}{a-b}\).
\[\frac{{b}^{2}}{{a}^{2}+ab+{b}^{2}}+\frac{ab(4a-b)}{(b-a)({b}^{2}+ba+{a}^{2})}+\frac{3{a}^{2}}{a-b}-2+4a\times 3\]
Simplify \(4a\times 3\) to \(12a\).
\[\frac{{b}^{2}}{{a}^{2}+ab+{b}^{2}}+\frac{ab(4a-b)}{(b-a)({b}^{2}+ba+{a}^{2})}+\frac{3{a}^{2}}{a-b}-2+12a\]
Simplify.
\[\frac{{b}^{2}}{{a}^{2}+ab+{b}^{2}}+\frac{ab(4a-b)}{(b-a)({b}^{2}+ab+{a}^{2})}+\frac{3{a}^{2}}{a-b}-2+12a\]
Simplify.
\[\frac{{b}^{2}}{ab+{a}^{2}+{b}^{2}}+\frac{ab(4a-b)}{(b-a)(ab+{a}^{2}+{b}^{2})}+\frac{3{a}^{2}}{a-b}-2+12a\]
b^2/(a*b+a^2+b^2)+(a*b*(4*a-b))/((b-a)*(a*b+a^2+b^2))+(3*a^2)/(a-b)-2+12*a
