Regional Competition

The claimed inequality is equivalent to $2ab+4a+4b+8\ge 2cd$, which can be written as $$2ab+4a+4b+a^2+b^2+c^2+d^2+4\ge 2cd$$ on account of the condition $a^2+b^2+c^2+d^2=4$. By the identity $$a^2+b^2+2ab+4a+4b+4=(a+b+2{)}^2$$ we arrive at the equivalent and obvious inequality $$(a+b+2{)}^2+(c-d{)}^2\ge 0$$ The case of equality occurs for $$a+b=-2\text{and}c=d$$ together with $a^2+b^2+c^2+d^2=4$. For instance when $a=b=c=d=-1$.