I am trying to figure out the following problem:

Find an upper bound on |z^{3}-2z^{2}+6z+2| defined by |z|<2.

I've done the following:

|z^{3}-2z^{2}+6z+2|

= |(x+iy)^{3}-2(x+iy)^{2}+6(x+iy)+2|

= |x^{3}+3ix^{2}y-3xy^{2}-iy^{3}-2(x^{2}+2ixy-y^{2})+6x+6iy+2|

= |x^{3}+3ix^{2}y-3xy^{2}-iy^{3}-2x^{2}-4ixy+2y^{2}+6x+6iy+2|

= |x^{3}-3xy^{2}-2x^{2}+2y^{2}+6x+2+(3x^{2}y-y^{3}-4xy+6y)i|

= √(x^{3}-3xy^{2}-2x^{2}+2y^{2}+6x+2)^{2}+ (3x^{2}y-y^{3}-4xy+6y)^{2 }I don't think I'm going this the right way. Any ideas? Should I not change to x+iy form?