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Concise higher mathematics solution
Let u = x+y and v = xy, then:

f(x+y,xy)

=f(u,v)

=2x^2 + 3xy + 2y^2 + 6

=2(x^2 + 2xy + y^2) - xy + 6

=2(x+y)^2 - xy + 6

=2u^2 - v + 6

As you can see, if the f(u, v) function equation uses X and Y to represent two independent variables, then its expression will become:

f(x,y) = 2x^2 - y + 6