\(1,H=a^3+b^3+3ab\left(a^2+b^2\right)+6a^2b^2\left(a+b\right)\)
\(=\left(a+b\right)\left(a^2-ab+b^2\right)+3ab[\left(a+b\right)^2-2ab]+6a^2b^2\left(a+b\right)\)
\(=\left(a+b\right)[\left(a+b\right)^2-3ab]+3ab[\left(a+b\right)^2-2ab]+6a^2b^2\left(a+b\right)\)
\(=1-ab+3ab\left(1-2ab+6a^2b^2\right)\)
\(=1-3ab+3ab-6a^2b^2+6a^2b^2\)
\(=1\)