Find the minimum value of this expression.

Find the minimum value of this expression.

Let $x$ and $y$ be two real numbers satisfying $$(x+y)^3 +4xy \geqslant
2.$$ Find the minimum value of the expression $$E=x^3+y^3 -2(x^2+y^2)-1.$$
I tried. We have $$2\leqslant (x+y)^3+(x+y)^2.$$ Therefore, $$x+y
\geqslant 1.$$ $$E = (x+y)^3-3xy(x+y)-2[(x+y)^2-2xy]-1.$$ Or $$E =
(x+y)^3-xy(3x+3y-4)-2(x+y)^2-1.$$ Using $$xy\leqslant \dfrac{(x+y)^2}{4}$$
and if $$3x + 3y -4 >0,$$ $$E \geqslant
(x+y)^3-\dfrac{(x+y)^2}{4}(3x+3y-4).$$ Put $t = x + y$, and using
derivative.
A problem is, if $$3x + 3y -4 <0,$$ how to prove?