Sorry I'm not good with the latex stuff yet, but can someone please solve this equation:
(2x+4y-2)dx + (-3x-2y+1)dy=0
Introduce new variables $\displaystyle (r,s)$ as to eliminate the $\displaystyle -2$ and $\displaystyle 1$ in your equation. So if $\displaystyle x = r + a$ and $\displaystyle y = s + b$ ($\displaystyle a$ and $\displaystyle b$ to be determined) then
$\displaystyle
\left(2(r+a) + 4(s+b) - 2 \right)\,dr + \left(-3(r+a) - 2(s+b) + 1 \right)\,ds =0
$
This means solving
$\displaystyle
\begin{array}{c}
2a + 4b = 2,\\
-3a -2b = -1
\end{array}$
Then your ODE becomes
$\displaystyle
\left(2r + 4s \right)\,dr + \left(-3 r - 2 s \right)\,ds =0
$
which is homogeneous.