Finding maximum power of resistor in 2 voltage source circuit
Giving some hints since its been hours...

Continuing what @efox29 was saying, you could write two current loop equations (KCL). 

Current one (\$i_{1}\$) going clockwise around the left loop and current two (\$i_2\$) going clockwise around the left loop, yields the following equations:

Left Loop


   \$6 -  10i_1 - 12 - 25i_1 - 15i_1 + 25i_2 = 0\$ Equation (1)


and Right Loop


   \$12 - R_Li_2 - 25i_2 + 25i_1 = 0\$ Equation (2)


Solve Equation 2 for \$i_1\$; substitute into Equation 1; solve for \$i_2\$ in terms or \$R_L\$; \$P = I_2^2R_L\$; solve for a Maximum.

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You can read about Thevenin's Theorem here.

Basically, it states that to find \$R_{Thevenin}\$ you short all voltage sources and open all current sources, then find the resistance looking in where the load was removed. That should make it very simple to get \$R_{Thevenin}\$

To find \$E_{Thevenin}\$ you remove \$R_L\$ and find the voltage at those terminals. You can use whatever method you prefer to solve for the voltage. As before you could write a KCL for the one loop. Solve for \$I\$ and find the voltage.

Draw your new circuit find the maximum power.