Homogenization, what it is in inequalities and how to utilize it to its fullest.

By AM-GM $$\prod_{cyc}a^a\leq\sum_{cyc}a^2.$$ Thus, it's enough to prove that $$(a+b+c+d)^3>(a+2b+3c+4d)(a^2+b^2+c^2+d^2),$$ which is obvious after full expanding.

Note that $a+2b+3c+4d \leqslant a + 3(b+c+d).$ Using the weighted AM-GM, we have $$a^2 + b^2 + c^2 + d^2 \geqslant a^ab^bc^cd^d.$$ We need to show that $$\left[a+3(b+c+d)\right](a^2 + b^2 + c^2 + d^2) \leqslant (a+b+c+d)^3,$$ equivalent to $$2(b+c+d)\left[a(b+c+d)-b^2-c^2-d^2\right]+2(a+b+c+d)(bc+cd+db) \geqslant 0.$$ Which is true because $a \geqslant b \geqslant c \geqslant d.$