\begin{equation*}
\begin{split}
(xy + 5r)^2& = (xy)^2 +2\cdot xy \cdot 5r + (5r)^2\\\\
& = x^2y^2 + 10xyr + 25r^2
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
(2mn \:-\: 8p)^2& = (2mn)^2\: -\; 2\cdot 2mn \cdot 8p + (8p)^2\\\\
& = 2^2m^2n^2 \:-\:32mnp + 8^2p^2\\\\
& = 4m^2n^2 \:-\:32mnp + 64p^2
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
(x^2 + y)^2& = (x^2)^2 + 2\cdot x^2 \cdot y + y^2\\\\
& = x^4 + 2x^2y + y^2
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
x^2 + 16xy + 64y^2& = x^2 + 2\cdot x \cdot 8y + (8y)^2\\\\
& = (x + 8y)^2
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
9x^2 – 18xy + 9y^2& = (3x)^2 \:-\: 2\cdot 3x \cdot 3y + (3y)^2\\\\
& = (3x \:-\:3y)^2
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
9m^2n^2 \:-\: 30mnr + 25r^2& = (3mn)^2\: – \:2\cdot 3mn \cdot 5r + (5r)^2\\\\
& = (3mn\:-\:5r)^2
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
(-x^3 \:-\:  y^5)^2& = [-(x^3 + y^5)]^2\:\:\:\:\:\color{blue} (-1)^2 = 1\\\\
& = (x^3 + y^5)^2\\\\
& = (x^3)^2 + 2\cdot x^3 \cdot y^5 + (y^5)^2\:\:\:\:\:\color{blue} (a^m)^n = a^{m\times n}\\\\
& = x^6 + 2x^3y^5 + y^{10}
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
(4m^7 \:-\: 5n^3)^2& = (4m^7)^2\: – \:2\cdot 4m^7 \cdot 5n^3 + (5n^3)^2\\\\
& = 4^2m^{14}\:-\: 40m^7n^3 + 5^2n^6\\\\
& = 16m^2n^{14} \:-\:40m^7n^3 + 25n^6
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
(\dfrac{3}{5}x^3 \:-\: \dfrac{5}{9}y^3)^2& = \left(\dfrac{3}{5}x^3\right)^2 \:- \: 2\cdot \dfrac{3}{5}x^3 \cdot \dfrac{5}{9}y^3 + \left(\dfrac{5}{9}y^3\right)^2\\\\
& = \dfrac{3^2}{5^2}x^6 \: – \:2\cdot \dfrac{\cancel{3}}{\cancel{5}} \cdot \dfrac{\cancel{5}}{\cancelto{3}{9}}x^3y^3 + \dfrac{5^2}{9^2}y^6\\\\
& = \dfrac{9}{25}x^6\: – \:\dfrac{2}{3}x^3y^3 + \dfrac{25}{81}y^6
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
\dfrac{2}{x^2} \: – \:2 + \dfrac{x^2}{2}& = \left(\dfrac{\sqrt{2}}{x}\right)^2\: – \: 2\cdot \cancel{\dfrac{\sqrt{2}}{x}}\cdot \cancel{\dfrac {x}{\sqrt{2}}} + \left(\dfrac{x}{\sqrt{2}}\right)^2\\\\
& = \left(\dfrac{\sqrt{2}}{x} \: – \: \dfrac{x}{\sqrt{2}}\right)^2\\\\
& = \left(\dfrac{\sqrt{2}}{x}\: – \: \dfrac{x}{\sqrt{2}}\times \color{blue} \dfrac{\sqrt{2}}{\sqrt{2}}\right)^2\\\\
& = \left(\dfrac{\sqrt{2}}{x} \: – \: \dfrac{1}{2}x\sqrt{2}\right)^2\\\\
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
\dfrac{m^6}{n^2} + 2m^2n^2 + \dfrac{n^6}{m^2}& = \left(\dfrac{m^3}{n}\right)^2 + 2\cdot \dfrac{\cancelto{\color{red}m^2\color{black}}{m^3}}{\cancel{n}} \cdot \dfrac{\cancelto{\color{red}n^2\color{black}}{n^3}}{\cancel{m}} + \left(\dfrac{n^3}{m}\right)^2\\\\
& = \left(\dfrac{m^3}{n} + \dfrac{n^3}{m}\right)^2
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
\dfrac{25m^4n^2}{3}\:-\: 2n + \dfrac{3}{25m^4}& = \left(\dfrac{5m^2n}{\sqrt{3}}\right)^2\: – \:2\cdot \dfrac{\cancel{5m^2}n}{\cancel{\sqrt{3}}}\cdot \dfrac{\cancel{\sqrt{3}}}{\cancel{5m^2}} + \left(\dfrac{\sqrt{3}}{5m^2}\right)^2\\\\
& = \left(\dfrac{5m^2n}{\sqrt{3}}\: – \:\dfrac{\sqrt{3}}{5m^2}\right)^2\\\\
& = \left(\dfrac{5m^2n}{\sqrt{3}}\times \color{blue} \dfrac{\sqrt{3}}{\sqrt{3}} \color{black} – \dfrac{\sqrt{3}}{5m^2}\right)^2\\\\
& = \left(\dfrac{5m^2n\sqrt{3}}{3} \: – \:\dfrac{\sqrt{3}}{5m^2}\right)^2
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
\left(\sqrt{2}m^2n\: -\: \dfrac{1}{mn^2}\right)^2& = \left(\sqrt{2}m^2n \right)^2 \: – \:2\cdot \sqrt{2}m^2n \cdot \dfrac{1}{mn^2} + \left(\dfrac{1}{mn^2}\right)^2\\\\
& = 2m^4n^2\: – \:\dfrac{2\sqrt{2}m}{n} + \dfrac{1}{m^2n^4}
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
\left(\dfrac{4x^2}{y} + \dfrac{y^2}{4x}\right)^2& = \left(\dfrac{4x^2}{y}\right)^2 + 2 \cdot \dfrac{4x^2}{y}\cdot \dfrac{y^2}{4x} + \left(\dfrac{y^2}{4x}\right)^2\\\\
& = \dfrac{16x^4}{y^2} + 2\cdot \dfrac{\cancelto{\color{red} x \color{black}}{4x^2}}{\cancel{y}} \cdot \dfrac{\cancelto{\color{red} y \color{black}}{y^2}}{\cancel{4x}} + \dfrac{y^4}{16x^2}\\\\
& = \dfrac{16x^4}{y^2} + 2xy + \dfrac{y^4}{16x^2}
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
(\sqrt[3]{x} \: – \: \sqrt[3]{y})^2& = (x^{\frac{1}{3}})^2 \: – \: 2\cdot x^{\frac{1}{3}} \cdot y^{\frac{1}{3}} + (y^{\frac{1}{3}})^2\\\\
& = x^{\frac{2}{3}}\: – \: 2\cdot (xy)^{\frac{1}{3}} + y^{\frac{2}{3}}\\\\
& = \sqrt[3]{x^2} \: – \: 2 \sqrt[3]{xy} + \sqrt[3]{y^2}
\end{split}
\end{equation*}