Problem:
Given the equation $9 x^2+5 y^2+8 y+5=0$, complete the square.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Complete the square}: \\
 5 y^2+8 y+\left(9 x^2+5\right)=0 \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{Subtract }5 \text{from }\text{both }\text{sides}: \\
 5 y^2+8 y+9 x^2=-5 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Group }\text{terms }\text{together }\text{on }\text{the }\text{left }\text{hand }\text{side, }\text{with }\text{a }\text{placeholder }\text{constant}: \\
 \left(5 y^2+8 y+\underline{\text{   }}\right)+9 x^2=\underline{\text{   }}-5 \\
\end{array}
 \\
 
\begin{array}{l}
 \left(5 y^2+8 y+\underline{\text{   }}\right)=5 \left(y^2+\frac{8 y}{5}+\underline{\text{   }}\right): \\
 \fbox{$5 \left(y^2+\frac{8 y}{5}+\underline{\text{   }}\right)$}+9 x^2=\underline{\text{   }}-5 \\
\end{array}
 \\
 
\begin{array}{l}
 
\begin{array}{l}
 \text{Take }\text{one }\text{half }\text{of }\text{the }\text{coefficient }\text{of }y \text{and }\text{square }\text{it. }\text{Then }\text{add }\text{it }\text{to }\text{both }\text{sides }\text{of }\text{the }\text{equation, }\text{multiplying }\text{by }\text{the }\text{factored }\text{constant }5 \text{on }\text{the }\text{right.} \\
 \text{Insert }\left(\frac{\frac{8}{5}}{2}\right)^2=\frac{16}{25} \text{on }\text{the }\text{left }\text{and }5\times \frac{16}{25}=\frac{16}{5} \text{on }\text{the }\text{right}: \\
\end{array}
 \\
 5 \left(y^2+\frac{8 y}{5}+\fbox{$\frac{16}{25}$}\right)+9 x^2=\fbox{$\frac{16}{5}$}-5 \\
\end{array}
 \\
 
\begin{array}{l}
 \frac{16}{5}-5=-\frac{9}{5}: \\
 5 \left(y^2+\frac{8 y}{5}+\frac{16}{25}\right)+9 x^2=\fbox{$-\frac{9}{5}$} \\
\end{array}
 \\
 
\begin{array}{l}
 y^2+\frac{8 y}{5}+\frac{16}{25}=\left(y+\frac{4}{5}\right)^2: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & 5 \fbox{$\left(y+\frac{4}{5}\right)^2$}+9 x^2=-\frac{9}{5} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/algebra/complete_square_w_steps/1381.txt'}