graph{(x^2+y^2+y)sqrt(x^2+y^2)=4x^2 [-8.046, 8.046, -4.02, 4.024]}
From the given
#r=-sin theta+4*cos^2 theta#
the conversion equations:
#r=sqrt(x^2+y^2)# and #theta=tan^-1(y/x)#
#tan theta=y/x# and # sin theta=y/sqrt(x^2+y^2)# and #cos theta=x/sqrt(x^2+y^2)#
Convert now, the given equation becomes
#sqrt(x^2+y^2)=-y/sqrt(x^2+y^2)+(4*x^2)/(x^2+y^2)#
multiply both sides of the equation by #(x^2+y^2)#
#(x^2+y^2)sqrt(x^2+y^2)=-(y*(x^2+y^2))/sqrt(x^2+y^2)+((x^2+y^2)(4*x^2))/(x^2+y^2)#
#(x^2+y^2)sqrt(x^2+y^2)=-(ycancel((x^2+y^2)))/cancelsqrt(x^2+y^2)+(cancel((x^2+y^2))(4*x^2))/cancel(x^2+y^2)#
#(x^2+y^2)sqrt(x^2+y^2)=-y*sqrt(x^2+y^2)+4*x^2#
#(x^2+y^2)sqrt(x^2+y^2)+y*sqrt(x^2+y^2)=4*x^2#
#(x^2+y^2+y)*sqrt(x^2+y^2)=4*x^2#
