= Solution
The parameter range is $0\leq\theta\leq1$. Omitting data-only constants from the <multinomial distribution> <likelihood>, its log is
$$
\ell(\theta)=(x_1+x_2)\log(1-\theta)+x_3\log\theta
+x_4\log(7+4\theta)+\mathrm{constant}.
$$
For an interior maximizer, the <likelihood> score must vanish:
$$
-\frac{x_1+x_2}{1-\theta}+\frac{x_3}{\theta}
+\frac{4x_4}{7+4\theta}=0.
$$
Multiply by $\theta(1-\theta)(7+4\theta)$ and collect powers to obtain
$$
\boxed{-4(x_1+x_2+x_3+x_4)\theta^2
+(-7x_1-7x_2-3x_3+4x_4)\theta+7x_3=0.}
$$
The <likelihood> is <concave>: its second <derivative> in the interior is
$$
\ell''(\theta)=-\frac{x_1+x_2}{(1-\theta)^2}
-\frac{x_3}{\theta^2}-\frac{16x_4}{(7+4\theta)^2}<0.
$$
Therefore an admissible interior root is the unique maximum. Endpoints must also be handled if some counts vanish. If $x_3>0$ and $x_1+x_2>0$, both endpoints have zero <likelihood> and the root is interior. If the maximum is at zero then $x_3=0$, and zero also satisfies the displayed polynomial; a maximum at one requires $x_1+x_2=0$, and one likewise satisfies it. Multiplying the score can introduce other endpoint roots, so not every polynomial root is an MLE: evaluate the actual <likelihood> on the feasible candidates.
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