Change of variables in a quadratic form

New variables. $x=u+2v$, $y=v$.

Direct substitution.
$$Q=(u+2v)^2+4(u+2v)v+3v^2=u^2+4uv+4v^2+4uv+8v^2+3v^2=u^2+8uv+15v^2.$$

Matrix form. The new matrix is $P^TAP$:
$$P^TAP=\begin{pmatrix}1&0\\2&1\end{pmatrix}\begin{pmatrix}1&2\\2&3\end{pmatrix}\begin{pmatrix}1&2\\0&1\end{pmatrix}.$$
Compute: $\begin{pmatrix}1&2\\2&3\end{pmatrix}\begin{pmatrix}1&2\\0&1\end{pmatrix}=\begin{pmatrix}1&4\\2&7\end{pmatrix}$. Then
$$\begin{pmatrix}1&0\\2&1\end{pmatrix}\begin{pmatrix}1&4\\2&7\end{pmatrix}=\begin{pmatrix}1&4\\4&15\end{pmatrix}.$$

Verification. $U^T\begin{pmatrix}1&4\\4&15\end{pmatrix}U=u^2+8uv+15v^2$. Consistent.

Conclusion. In $(u,v)$, $Q=u^2+8uv+15v^2$.