The equation of a circle is given in general form. Complete parts a through d to follow the step by step process to sketch the graph of the circle. \( x^{2}+y^{2}+4 x-8 y+4=0 \) a) Write the equation of the circle in standard form. The equation is (Simplify your answer.)

To convert the given equation of the circle from general form to standard form, we need to complete the square for both the \(x\) and \(y\) terms. The given equation is: \[ x^{2} + y^{2} + 4x - 8y + 4 = 0 \] First, we can rearrange the equation by moving the constant term to the other side: \[ x^{2} + 4x + y^{2} - 8y = -4 \] Next, we will complete the square for the \(x\) terms and the \(y\) terms. 1. **Completing the square for \(x\):** - Take the coefficient of \(x\) (which is 4), divide it by 2 to get 2, and then square it to get \(2^2 = 4\). - Add and subtract 4 inside the equation. 2. **Completing the square for \(y\):** - Take the coefficient of \(y\) (which is -8), divide it by 2 to get -4, and then square it to get \((-4)^2 = 16\). - Add and subtract 16 inside the equation. Now, we rewrite the equation: \[ (x^{2} + 4x + 4) + (y^{2} - 8y + 16) = -4 + 4 + 16 \] This simplifies to: \[ (x + 2)^{2} + (y - 4)^{2} = 16 \] Now, we have the equation in standard form: \[ (x + 2)^{2} + (y - 4)^{2} = 16 \] This represents a circle with center \((-2, 4)\) and radius \(4\) (since \( \sqrt{16} = 4\)). So, the final answer for part a is: \[ (x + 2)^{2} + (y - 4)^{2} = 16 \]