equation of the form P (θ) = 0, where P is a polynomial, satisfied by the maximum likelihood estimator

A multinomial model with a sample of size n, let X = (X1, X2, X3, X4). And probability: P1=(1/2 +θ/4), P2=(1/4 − θ/4) P3:(1/4 − θ/4) P4:(θ/4). Where the parameter θ satisfies 0 ≤ θ ≤ 1.

How can I obtain an equation of the form P (θ) = 0, where P is a polynomial, satisfied by the maximum likelihood estimator?

I think I need to use Lagrange multipliers:
L(p1, p2, …, pm, λ) = l(p1, p2, …, pm) + λ(1 − ? pi) but I dont know how to obtain an equation of the form P (θ) = 0