Let a, b, c, d be positive integers such that $a\ge b\ge c\ge d,$ then  which of the following can be  correct about the  roots of the equation  $x^4-a{x}^3-b{x}^2-cx-d=0$

Suppose that m is an integer root of $x^4-a{x}^3-b{x}^2-cx-d=0$ . As  $d\ne 0$, we have $m\ne 0$ .Suppose now that  $\text{m}>0$.Then ${\text{m}}^{\text{4}}–{\text{am}}^{\text{3}}={\text{bm}}^{\text{2}}+\text{cm }+\text{d }>0$ and hence  $\text{m}>\text{a}\ge d$  .On the other hand $\text{d}=\text{m}({\text{m}}^{\text{3}}–{\text{am}}^{\text{2}}–\text{bm}–\text{c})$ and hence m divides d, so ,a    $\text{m}\le d$    contradiction.
If $\text{m}<0$ , then writing $\text{n}=–\text{m}>0$ we have  ${\text{n}}^{\text{4}}+{\text{an}}^{\text{3}}–{\text{bn}}^{\text{2}}+\text{cn}–\text{d }={\text{n}}^{\text{4}}+{\text{n}}^{\text{2}}(\text{an}–\text{b})+(\text{cn}–\text{d})>0$ , a contradiction. This proves  that the given polynomial has no integer roots.