\begin{equation*}
\begin{split}
(x + y + 7)^2& = x^2 + y^2 + 7^2 + 2\cdot x \cdot y + 2\cdot x \cdot 7 + 2\cdot y \cdot 7\\\\
& = x^2 + y^2 + 49 + 2xy + 14x + 14y
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
(x + 2y\: – \:3)^2& = x^2 + (2y)^2 + (-3)^2 + 2\cdot x \cdot 2y + 2\cdot x \cdot (-3) + 2\cdot 2y \cdot (-3)\\\\
& = x^2 + 4y^2 + 9 + 4xy -6x – 12y
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
(2m \:-\:n\:-\:5r)^2& = (2m)^2 + (-n)^2 + (-5r)^2 + 2\cdot 2m \cdot (-n) + 2\cdot 2m \cdot (-5r) + 2\cdot (-n) \cdot (-5r)\\\\
& = 4m^2 + n^2 + 25r^2 – 4mn – 20mr + 10nr
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
9x^2 + y^2 + 16 + 6xy + 24x + 8y& = (3x)^2 + y^2 + 4^2 + 2\cdot 3x \cdot y + 2\cdot 3x \cdot 4 + 2\cdot y\cdot 4\\\\
& = (3x + y + 4)^2
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
1 + 9x^2 + 25y^2 + 6x \:- \:10y\: – \:30 xy& = 1^2 + (3x)^2 + (-5y)^2 + 2\cdot 1 \cdot 3x + 2 \cdot 1 \cdot (-5y) + 2\cdot 3x \cdot (-5y)\\\\
& = (1 + 3x – 5y)^2
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
a^2 + 4b^2 + 9c^2\:-\:4ab \:-\:6ac + 12bc& = a^2 + (-2b)^2 + (-3c)^2 + 2\cdot a \cdot (-2b) + 2\cdot a \cdot (-3c) + 2\cdot (-2b)\cdot (-3c)\\\\
& = (a \: – \: 2b \: – \: 3c)^2
\end{split}
\end{equation*}