\((a+b):=(a+b)(a+b)^{2}\)
\(=\left(a^{+}+b\right)\left(a^{2}+2 a b+b^{2}\right)=a^{3}+b^{3}+3 a^{2} b+3 a b^{2}\)
we factorise out \(3 a b\) and transfer to the other side of the equality sign i.e.
\(\Rightarrow(a+b)^{i}-3 a b(a+b)=a^{3}+b^{3}\)
but \((a+b):=(a+b)\left(a^{2}+2 a b+b^{2}\right)\)
\(a^{2}+b^{2}=(a+b)\left(a^{2}+2 a b+b^{2}\right)-3 a b(a+b)\)
\(a^{2}+b^{2}=(a+b)\left[a^{2}+b^{2}+2 a b-3 a b\right]\)