x²+y²+y = 0

For all problems changing between rectangular and polar form,
you should draw this right triangle:

 

to replace all x's and y's by r's and q's. Also when you see "x²+y²" 
you can immediately replace the TWO terms by the ONE term r². 

So from that triangle and the Pythagorean theorem, you can easily see
that x²+y² = r², so substituting the ONE TERM r² for the first TWO
terms, you have

r²+y = 0

Then you see from that right triangle that y/r = sin(q) and y = r∙sin(q).
So the final polar equation is

r²+r∙sin(q) = 0

Now to solve that for r:

r²+r∙sin(q) = 0

r[r+sin(q)] = 0

r=0; r+sin(q) = 0
     r = -sin(q)   
 

We can ignore r=0 which is the equation of the origin, but since 
the circle goes through the origin.  So the answer is the polar 
equation:

r = -sin(q

Edwin