Decide what numbers must be added to both sides of the equation to make the left side the sum of a perfect square trinomial in \( x^{2}+24 x+y^{2}-20 y=-4 \) The number 144 should be added to both sides of the equation to make the left side the sum of a perfect square trinomial in \( x \). (Simplify your answer.) The number \( \square \) should be added to both sides of the equation to make the left side the sum of a perfect square trinomial in \( y \). (Simplify your answer.)

To transform the left side of the equation \( x^{2} + 24x + y^{2} - 20y = -4 \) into the sum of perfect square trinomials, we need to complete the square for both the \( x \) and \( y \) terms. ### Completing the square for \( x \): The expression involving \( x \) is \( x^{2} + 24x \). 1. Take half of the coefficient of \( x \) (which is 24), square it: \[ \left(\frac{24}{2}\right)^{2} = 12^{2} = 144 \] 2. We need to add 144 to both sides of the equation. ### Completing the square for \( y \): The expression involving \( y \) is \( y^{2} - 20y \). 1. Take half of the coefficient of \( y \) (which is -20), square it: \[ \left(\frac{-20}{2}\right)^{2} = (-10)^{2} = 100 \] 2. We need to add 100 to both sides of the equation. ### Summary: - The number \( 144 \) should be added to both sides to complete the square for \( x \). - The number \( 100 \) should be added to both sides to complete the square for \( y \). Thus, the final answer is: - The number \( 100 \) should be added to both sides of the equation to make the left side the sum of a perfect square trinomial in \( y \).