Solve for the roots in simplest form by completing the square: \[ -5 x^{2}-10 x-50=0 \]

To solve the quadratic equation by completing the square: \[ -5x^2 - 10x - 50 = 0 \] **Step 1: Simplify the Equation** First, divide the entire equation by \(-5\) to make the coefficient of \(x^2\) positive: \[ x^2 + 2x + 10 = 0 \] **Step 2: Move the Constant Term** Subtract \(10\) from both sides to isolate the quadratic and linear terms: \[ x^2 + 2x = -10 \] **Step 3: Complete the Square** To complete the square, take half of the coefficient of \(x\), which is \(1\), and square it to get \(1\). Add this to both sides: \[ x^2 + 2x + 1 = -10 + 1 \] \[ (x + 1)^2 = -9 \] **Step 4: Solve for \(x\)** Take the square root of both sides, remembering to include both the positive and negative roots: \[ x + 1 = \pm \sqrt{-9} \] \[ x + 1 = \pm 3i \] Subtract \(1\) from both sides to solve for \(x\): \[ x = -1 \pm 3i \] **Final Answer:** The solutions are \(x = -1 + 3i\) and \(x = -1 - 3i\).