An algebra problem by Aly Ahmed

The given expression is $(a+\dfrac{1}{a}+1)(b+\dfrac{1}{b}+1)(c+\dfrac{1}{c}+1)(d+\dfrac{1}{d}+1)\geq (2+1)(2+1)(2+1)(2+1)=\boxed {81}$

$\begin{aligned} \frac {(a^2+a+1)(b^2+b+1)(c^2+c+1)(d^2+d+1)}{abcd} & = \left(a+1+\frac 1a\right)\left(b+1+\frac 1b\right)\left(c+1+\frac 1c\right)\left(d+1+\frac 1d\right)\end{aligned}$

By AM-GM inequality , for positive real, $a + \dfrac 1a \ge 2\sqrt{a \times \dfrac 1a} = 2$ . Therefore,

$\begin{aligned} \left(a+1+\frac 1a\right)\left(b+1+\frac 1b\right)\left(c+1+\frac 1c\right)\left(d+1+\frac 1d\right) \ge 3^4 = \boxed{81}\end{aligned}$