Hello! My solution: $\prod_{cyc}(16a^2+8b+17) = \prod_{cyc}(16a^2 + 1+ 8b+16) \geq \prod_{cyc}(8a + 8b + 16) \geq 2^9\prod_{cyc}((a + 1)+(b+1)) \geq $ $\geq 2^9\prod_{cyc}(2\sqrt{(a+1)(b+1)}) = 2^{12}\prod_{cyc}(a+1)$ Equality we have for $a=b=c=1/4$.