Finding the thevenin voltage
You have to find the Thevenin equivalent circuit (TEC) of the circuit without the load, i.e. the part of the circuit left of the two terminals (marked as two empty circles) the load is connected to.

In order to find the TEC of that circuit find e.g. by nodal (or mesh) analysis


the open circuit voltage \$v_{oc}\$ (i.e voltage between the unconnected terminals) 

and
the short circuit current \$i_{sc}\$ (i.e. current through a short between the terminals)



of the circuit.

Then the TEC is a voltage source of \$v_{th} = v_{oc}\$ with series resistance \$R_{th} = \frac{v_{oc}}{i_{sc}}\$:
 

You probably know that maximum power transfer is happening if \$R_L = R_{th}\$.
Therefore the voltage across \$R_L\$ will be \$\frac{1}{2}v_{th}\$ so the power dissipated in \$R_L\$ is \$P = \frac{(\frac{1}{2}v_{th})^2}{R_L}\$.