Area enclosed by curve $f(x,y)$ defined implicitly Find area bounded by the curve 
  $$5x^2+2y^2+6xy+7x+6y+6=0$$


I can find the area using integration for curves defined explicitly in $x$ or $y$. I have no idea how to do this.
The hint.

By the affine transformation with  determinant $\Delta$ write the equation of this ellipse in the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the needed area is equal to $\frac{\pi ab}{\Delta}$

I got the following.
$$\left(2x+y+\frac{1}{2}\right)^2+\left(x+y+\frac{5}{2}\right)^2=\frac{1}{2}.$$
Can you end it now?
just like Michael Rozenberg said, you can use the completing the square approach to make the equation take the form $a^{2}x^{2}+b^{2}y^{2}=a^{2}b^{2}$ the the result follows.