Reducing a quadratic form

Completing the square.
$$Q(x,y) = x^2 + 4xy + 5y^2 = (x+2y)^2 + y^2.$$

Change of variables. Let $u = x+2y$ and $v = y$. Then $Q = u^2 + v^2$, which is a positive definite quadratic form.

Signature. The signature is $(2,0,0)$: two positive squares, no negative, no zero.

Remark. The associated symmetric matrix is $\begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix}$, whose eigenvalues are $3 \pm 2\sqrt2$, both positive, confirming positive definiteness.