I have been trying to find the smallest possible value of $x^2 + 4xy + 5y^2 - 4x - 6y + 7$, but I do not seem to have been heading in any direction which is going to give me an answer I feel certain is correct. Any hints on how to algebraically approach finding this value would be appreciated. I prefer not to be told what the value is.
Note that $x^2 + 4xy + 5y^2 - 4x - 6y + 7=(x+2y)^2+y^2-4x-6y+7$.
Let $u=x+2y$. Write our expression in terms of $u$ and $y$, and complete the squares.
Remark: The approach may have looked a little ad hoc, but the same basic idea will work for any quadratic polynomial $Q(x,y)$ that has a minimum or maximum. For quadratic polynomials $Q(x_1,x_2,\dots, x_n)$, one can do something similar, but it becomes useful to have more theory. You may want to look into the general diagonalization procedure.