All roots of the polynomial equation $x^4-4x^3+ax^2+bx+1=0$ are positive real numbers. Show that all the roots of the polynomial are equal.

Suppose that all roots of the polynomial equation
$$x^4-4x^3+ax^2+bx+1=0$$
are positive real numbers. Show that all the roots of the polynomial are equal.

My work:
I assume the contraposition that all the roots are not equal.
Assume that the roots are $\alpha,\beta,\gamma,\delta$
So,$\alpha+\beta+\gamma+\delta=4$
and,$\alpha\beta\gamma\delta=1$
Here, by observation I can see that this holds for all the roots to be equal to 1, but I cannot prove it. Please help!

Answer

Hint : you are then in the equality case of the AM-GM inequality