<s>[INST] Prove $\frac{a(b+c)}{b^2+bc+c^2} \geqslant \frac{8a^2+2a(b+c)-(b-c)^2}{3(a^2+b^2+c^2+ab+bc+ca)}.$ Let $a,\,b,\,c$ are non-negative such that $ab+bc+ca>0.$ Prove that
$$\frac{a(b+c)}{b^2+bc+c^2} \geqslant \frac{8a^2+2a(b+c)-(b-c)^2}{3(a^2+b^2+c^2+ab+bc+ca)}. \quad (1)$$
Note. Because
$$\sum \frac{8a^2+2a(b+c)-(b-c)^2}{3(a^2+b^2+c^2+ab+bc+ca)} = 2.$$
So, from $(1)$ we get know inequality of Darij Grinberg
$$\frac{a(b+c)}{b^2+bc+c^2}+\frac{b(c+a)}{c^2+ca+a^2}+\frac{c(a+b)}{a^2+ab+b^2} \geqslant 2.$$
My proof use sum of squares method.
 [/INST] We have$:$
$$48\, \left( b+c \right) ^{2} \left( {b}^{2}+bc+{c}^{2} \right) 
 \left( {a}^{2}+ab+ac+{b}^{2}+bc+{c}^{2} \right) 
\cdot (\text{LHS}-\text{RHS})$$
$$=48\,a ( b+c )  \Big[b^2+c^2-a(b+c)\Big] ^{2}+4\,(b-c)^2 \Big[ 2\, a(b+c)-(2b^2+bc+2c^2) \Big] ^{2}$$
$$+4\,bc \left( 8\,{b}^{2}+7\,bc+8\,{c}^{2} \right)  \left( b-c \right) ^
{2} \geqq 0.$$
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