Let $a,b,c$ be reals Prove that$$(a^3+b^3+c^3+a^2b+b^2c+c^2a)^2\geq 4(a^2+b^2+c^2)(a^3b+b^3c+c^3a)$$

Phorphyrion wrote: \[(a^5+b^5+c^5+a^3c^2+b^3a^2+c^3b^2)^2\geq 4(a^2+b^2+c^2)(a^5b^3+b^5c^3+c^5{\color{red}a}^3)\] Maybe a Typo?

sqing wrote: Let $a,b,c$ be reals Prove that$$(a^3+b^3+c^3+a^2b+b^2c+c^2a)^2\geq 4(a^2+b^2+c^2)(a^3b+b^3c+c^3a)$$ WLOG $c\geq b$ and $c\geq a$. $$(a^3+b^3+c^3+a^2b+b^2c+c^2a)^2- 4(a^2+b^2+c^2)(a^3b+b^3c+c^3a)= (-a^3+b^3+c^3+a^2b-b^2c-c^2a)^2+4b^2c^2(c-b)(c-a)\geq 0$$

Phorphyrion wrote: The numbers $a$, $b$, and $c$ are real. Prove that $$(a^5+b^5+c^5+a^3c^2+b^3a^2+c^3b^2)^2\geq 4(a^2+b^2+c^2)(a^5b^3+b^5c^3+c^5a^3)$$ WLOG $c\geq b$ and $c\geq a$ $$(a^5+b^5+c^5+a^3c^2+b^3a^2+c^3b^2)^2- 4(a^2+b^2+c^2)(a^5b^3+b^5c^3+c^5a^3)=(a^5-b^5-c^5+a^3c^2-b^3a^2+c^3b^2)^2+4b^2c^2(c^3-b^3)(c^3-a^3)\geq 0$$