#x=rcos(theta)#
#y=rsin(theta)#
#r^2=x^2+y^2#
Make the necessary substitutions
#3rsin(theta)=2(rcos(theta))^2-2rcos(theta)rsin(theta)-rcos(theta)#
Simplify
#3rsin(theta)=2r^2cos^2(theta)-2r^2cos(theta)sin(theta)-rcos(theta)#
Add #rcos(theta)# to both sides
#3rsin(theta)+rcos(theta)=2r^2cos^2(theta)-2r^2cos(theta)sin(theta)#
Factor out #r# and #r^2#
#r(3sin(theta)+cos(theta))=r^2(2cos^2(theta)-2cos(theta)sin(theta))#
Isolated #r^2#
#(r(3sin(theta)+cos(theta)))/(2cos^2(theta)-2cos(theta)sin(theta))=(r^2cancel(2cos^2(theta)-2cos(theta)sin(theta)))/cancel(2cos^2(theta)-2cos(theta)sin(theta))#
#(r(3sin(theta)+cos(theta)))/(2cos^2(theta)-2cos(theta)sin(theta))=r^2#
Gather #r# to the right hand side
#((cancelr(3sin(theta)+cos(theta)))/(2cos^2(theta)-2cos(theta)sin(theta)))/cancelr=r^cancel2/cancelr#
Simplify
#(3sin(theta)+cos(theta))/(2cos^2(theta)-2cos(theta)sin(theta))=r#
Check out my tutorials on this subject.
