Question Solve \( 5 a^{2}+10 a=-205 \) by completing the square. (Use a comma to separate multiple values.)

To solve \( 5 a^{2} + 10 a = -205 \) by completing the square, first, divide everything by 5: \[ a^{2} + 2a = -41 \] Next, complete the square by taking half of the coefficient of \( a \) (which is 2), squaring it, and adding it to both sides: \[ a^{2} + 2a + 1 = -41 + 1 \] This simplifies to: \[ (a + 1)^{2} = -40 \] Taking the square root of both sides gives us: \[ a + 1 = \pm \sqrt{-40} \] Since \( \sqrt{-40} = \sqrt{40}i = 2\sqrt{10}i \), we have: \[ a + 1 = \pm 2\sqrt{10}i \] Finally, solving for \( a \): \[ a = -1 \pm 2\sqrt{10}i \] So, the solutions are \( a = -1 + 2\sqrt{10}i, -1 - 2\sqrt{10}i \).