Question

Find the equation in polar form: -7x^(2)-3xy-7y^(2)= -2

Answer 1

-7 x^2-3 x y-7 y^2+2 = 0

Answer 2

Simply put rcosθ in place of x & rsinθ in place of y & solve it
we have 7*r^2*(cosθ)^2+3*r*Sinθ*r*Cosθ+7*r^2*(Sinθ)^2=2
Now realising that (cosθ)^2+(Sinθ)^2=1 we have

7*r^2((cosθ)^2+(Sinθ)^2)+3*r^2*Sinθ*Cosθ=2

& r^2*(7+3*Sinθ*Cosθ)=2
&
r=sqrt(2/(7+3*Sinθ*Cosθ))

This is the answer.
More details about this curve is here
http://www.wolframalpha.com/input/?i=+-7x%5E%282%29-3xy-7y%5E%282%29%3D+-2