Number of integral solutions of polynomial

Question

Consider the equation $f(x) = x^4-ax^3-bx^2-cx-d=0,$ $a, b, c, d \in \mathbb Z^+,$ $ a≥b≥c≥d$ then number of integral solutions can be.

I am unable to use Integral Root theorem and just reached to the conclusion that if $\alpha$ is positive root of$f(x)$ then $\alpha >a$, nothing else. How should it be done?

Answer 1

Case 1

Let $x ≥ 1$ be an integral root of $f(x)$ $\implies d = x(x^3 - ax^2 - bx - c)$ $\implies d≥x$

Also

$x^3(x - a) = bx^2+ cx + d$ $\implies x > a$

That's a contradiction as $a≥d$

Case 2

Let $x ≤ -1$

Put $x = -y$

$y^4 + ay^3 - by^2+ cy - d = y^4+y^2(ay - b) + cn - d > 0$

Thus $f(x)$ has no integral roots.