Solve for the roots in simplest form by completing the square: \[ 4 x^{2}-32 x+244=0 \]

To solve the equation \( 4 x^{2} - 32 x + 244 = 0 \) by completing the square, let's start by simplifying it. 1. Divide the entire equation by 4: \[ x^2 - 8x + 61 = 0 \] 2. Move the constant term to the other side: \[ x^2 - 8x = -61 \] 3. To complete the square, take half of the coefficient of \( x \) (which is \(-8\)), square it, and add it to both sides. Half of \(-8\) is \(-4\), and squaring gives us \(16\): \[ x^2 - 8x + 16 = -61 + 16 \] \[ x^2 - 8x + 16 = -45 \] 4. Now, the left-hand side can be factored as a square: \[ (x - 4)^2 = -45 \] 5. Taking the square root of both sides: \[ x - 4 = \pm \sqrt{-45} \] \[ x - 4 = \pm \sqrt{45} i \] Since \(\sqrt{-1}\) is \(i\) and \(\sqrt{45} = 3\sqrt{5}\): \[ x - 4 = \pm 3\sqrt{5} i \] 6. Finally, add 4 to both sides to find \( x \): \[ x = 4 \pm 3\sqrt{5} i \] The roots of the equation \( 4 x^{2} - 32 x + 244 = 0 \) are: \[ x = 4 + 3\sqrt{5} i \quad \text{and} \quad x = 4 - 3\sqrt{5} i \]