This is not a circle equation. Center and radius cannot be determined.
Step 1: Move the loose number to the right side.
$$ x^2 + y^2 + 4 y = -21 $$Step 2: Write the equation in the following form.
$$ x^2 + \left( y^2 + 4 y + \color{blue}{\clubsuit} \right) = -21 + \color{blue}{\clubsuit} $$( the symbol $ \color{blue}{\clubsuit} $ , is the number needed to complete the square )
Step 3: To find $\color{blue}{\clubsuit}$ , take the y - term, divide it by 2 an then square the result.
$$ \color{blue}{\clubsuit} = \left( \frac{ 4 }{2} \right)^2 = 4 $$Step 4: Put steps 2 and 3 together.
$$ \begin{aligned} x^2 + \left( y^2 + 4 y + 4 \right) &= -21 + 4 \\ (x-0)^2 + \left( y + 2 \right)^2 &= -17 \end{aligned}$$Step 5: Since, the value on the right side is not positive we conclude that the starting equation do not represent a circle.