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\begin{array}{l}  x^2 + 4x = 0 \\  (x + 2)^2  - 4 = 0 \\  (x + 2)^2  = 4 \\  x + 2= -2 \vee x + 2 = 2 \\  x = - 4 \vee x = 0 \\  \end{array}


\begin{array}{l}  x^2  + 7x + 12\frac{1}{4} = 0 \\  \left( {x + 3\frac{1}{2}} \right)^2=0 \\  x + 3\frac{1}{2} = 0 \\  x =-3\frac{1}{2} \\  \end{array}


\begin{array}{l}  x^2  - 4x = 0 \\  (x - 2)^2  - 4 = 0 \\  (x - 2)^2  = 4 \\  x - 2 =-2 \vee x - 2 = 2 \\  x = 0 \vee x = 4 \\  \end{array}


\begin{array}{l}  x^2  = x \\  x^2  - x = 0 \\  \left( {x - \frac{1}{2}} \right)^2  - \frac{1}{4} = 0 \\  \left( {x - \frac{1}{2}} \right)^2  = \frac{1}{4} \\  x - \frac{1}{2} =  - \frac{1}{2} \vee x - \frac{1}{2} = \frac{1}{2} \\  x = 0 \vee x = 1 \\  \end{array}


\begin{array}{l}  x^2 + 8x + 4 = 0 \\ (x+4)^{2}-16+4=0 \\ (x+4)^{2}-12=0 \\ (x+4)^{2}=12 \\ x+4=-\sqrt{12}\vee x+4=\sqrt{12} \\ x=-4-\sqrt{12}\vee x=-4+\sqrt{12} \\ \end{array}


\begin{array}{l}  x^2  = 5x + 4 \\  x^2  - 5x = 4 \\  \left( {x - 2\frac{1}{2}} \right)^2  - 6\frac{1}{4} = 4 \\  \left( {x - 2\frac{1}{2}} \right)^2  = 10\frac{1}{4} \\  x - 2\frac{1}{2} =  - \sqrt {10\frac{1}{4}}  \vee x - 2\frac{1}{2} = \sqrt {10\frac{1}{4}}  \\  x = 2\frac{1}{2} - \frac{1}{2}\sqrt {41}  \vee x = 2\frac{1}{2} + \frac{1}{2}\sqrt {41}  \\  \end{array}


\begin{array}{l}  x^2  + 2x = 31 \\  (x + 1)^2  - 1 = 31 \\  (x + 1)^2  = 32 \\  x + 1 =  - \sqrt {32}  \vee x + 1 = \sqrt {32}  \\  x =  - 1 - 4\sqrt {2}  \vee x =  - 1 + 4\sqrt {2}  \\  \end{array}


\begin{array}{l}  2x^2  + 4x = 3 \\  2\left( {x^2  + 2x} \right) = 3 \\  2((x + 1)^2  - 1) = 3 \\  2(x + 1)^2  - 2 = 3 \\  2(x + 1)^2  = 5 \\  (x + 1)^2  = 2\frac{1}{2} \\  x + 1 =  - \sqrt {2\frac{1}{2}}  \vee x + 1 = \sqrt {2\frac{1}{2}}  \\  x =  - 1 - \frac{1}{2}\sqrt {10}  \vee x =  - 1 + \frac{1}{2}\sqrt {10}  \\  \end{array}


\begin{array}{l}  x^2  - 27 = 2x \\  x^2  - 2x - 27 = 0 \\  (x - 1)^2  - 28 = 0 \\  x = 1 - 2\sqrt {7}  \vee x = 1 + 2\sqrt {7}  \\  \end{array}


\begin{array}{l}  (x + 1)(x + 5) = 2x + 13 \\  x^2  + 6x + 5 = 2x + 13 \\  x^2  + 4x + 5 = 13 \\  (x + 2)^2  + 1 = 13 \\  (x + 2)^2  = 12 \\  x =  - 2 - 2\sqrt {3}  \vee x =  - 2 +2\sqrt {3}  \\  \end{array}

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