Answer a is not possible since it reduces to
$m = \frac{1}{2} 2m + \frac{3}{2}\frac{2m}{3} = 2m$
which has no sense. Idem for answer c, $m\neq\frac{1}{4}m$.
Remains answers b or d.
For answer b, we have
$u(2m,0)=8m$
and for answer d,
$u(0,\frac{2m}{3}) = \frac{28}{3}m = (9+\frac{1}{3})m$
As you can see, $ (9+\frac{1}{3}) > 8$.
In the general case, to find the maximum of your utility function given a monetary constrain, you can formalize and maximize the following Lagrangian function
$L(x_1,x_2,\lambda)=u(x_1,x_2)+\lambda(m-p_1 x_1 - p_2 x_2)$
where $p_1$ and $p_2$ are prices. In your case, those are $\frac{1}{2}$ and $\frac{3}{2}$ respectively.
Suite
As usual, the story behind equations is of first importance.
Remember that $x_1$ and $x_2$ are two perfect substitutes. This means that the individual will spend all her income either in $x_1$ or in $x_2$, and will chose the good which provides her with the highest utility. And this is actually the story that the use of the Lagrangian function tells you.
As you mentioned in your comment below, you get first order conditions which seem to be contradictory. But they are not. Indeed, the two goods will never be bought simultaneously. Which means that you will get either $\lambda = 8$ or $\lambda = \frac{28}{3}$ in a mutually exclusive manner.
Recall what $\lambda$ is : it is a shadow price. In other words, it expresses how much your objective function, $u(x_1,x_2)$, will increase, if your constrain, $m$, increases by $1$.
Thus, if $m$ increases by $1$, and the individual spends all her income in $x_1$, $u$ will increase by $8$.
If she spends all her income in $x_2$, $u$ will increase by $\frac{28}{3}$.
Which good will the individual chose ?