Langrange Multiplier Method

Need help using the Lagrange Multiplier Method on this problem. I could easily optimize this with direct substitution but I need to know how to solve it using this method.

$\displaystyle z=f(x,y)=3(x-5)^2+4y+1307$

The Function is constrained by $\displaystyle x+y=144$

so I have $\displaystyle L(x,y)=3(x-5)^2+4y+1307-lambda(x+y-144)$

Then $\displaystyle Lx(x,y)=6(x-5)-lambda=0$

$\displaystyle Ly(x,y)=4-lambda=0$

$\displaystyle g(x,y)=x+y-144=0$

am I going about this right?