I assume you mean that, given that $a,b,c,d$ are positive and, $$(1+a)(1+b)(1+c)(1+d)=16$$Find the largest value of $abcd$.

Let $ a,b,c,d $ be positive real numbers such that $ (a+1)(b+4)(c+4)(d+1) \leq 81 .$ Prove that$$abcd\leq 1$$Let $ a,b,c,d $ be positive real numbers such that $ (a+1)(b+2)(c+2)(d+1) \leq 17+12\sqrt 2 .$ Prove that$$abcd\leq 1$$