I'm sure you have some change of variables formulas in your PDE course, meant to simplify the given equation. I don't remember the formulas, but there is a way to avoid them. The equation can be written in the following way:
$ A u = \sin(t+x) $ where $A$ is the operator
$ A= 3\frac{\partial^2}{\partial t^2}+10\frac{\partial^2}{\partial t \partial x} +3 \frac{\partial^2}{\partial x^2}.$
In this case, the operator can be factored (just like a binomial expression) in the following way:
$ A =\left( \frac{\partial }{\partial t}+3\frac{\partial}{\partial x}\right)\left(3\frac{\partial}{\partial t}+\frac{\partial }{\partial x} \right).$
Now, change the variables such that each of the two factors will be a partial derivation.
$ \frac{\partial}{\partial y}=\left( \frac{\partial }{\partial t}+3\frac{\partial}{\partial x}\right), \frac{\partial}{\partial z}=\left(3\frac{\partial}{\partial t}+\frac{\partial }{\partial x} \right)$
For this, change the variables to $\displaystyle \begin{cases} t=y+3z \\ x=3y+z \end{cases}$ and define $w(y,z)=u(t,x)$. Then we have
$\frac{\partial^2w }{\partial y \partial z}=Au= \sin(x+t)=\sin(4y+4z)$
Integrate two times, with respect to $y,z$ and you will find $w$. From there it is easy to get to $u$.
This might not be the most efficient method (for an exam, for example), but it might show from where the change of variables formulas come from. I did this in my exam, once, because I didn't remember the formulas. (I got zero points, because my teacher didn't like the method and there were some small mistakes). It is best to learn all the cases for change of variable, because that gains you time in an exam, but is also important to know from where those change of variable formulas come from.