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Further Maths Question Help

Hi so in school a while back i did the below question where we had to find out z in terms of x+iy

z(2+i) = (1+2i) (1+2i)

This was my working out in school...
2z + zi = -3 + 4i
2(x+iy) + i(x+iy) = -3 + 4i
2x + 2iy + ix - y = -3 + 4i
(2x-y) i(2y+x) = -3 + 4i
2x-y= -3
2y+x = 4

And then my answers in the book were x= -2/5 and y=11/5
Have I done something completely wrong or could someone explain to me the step between finding the real and imaginary components in terms of y and x and how we actually figured those answers out.
Reply 1
It's just a pair of simultaneous equations, surely?

https://www.bbc.co.uk/bitesize/guides/z9y9jty/revision/1
Reply 2
Or multiply through by the conjugate of (2+i)
Reply 3
In other words, all that has happened here is that you have not quite finished.
Just solve the simultaneous equations.

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