We want to solve the equation -x² = 3(x + 1).

Step 1: Expand the right side of the equation:

-x² = 3(x + 1) = 3x + 3

Step 2: Move all terms to one side to form a quadratic equation:

-x² – 3x – 3 = 0

Step 3: Identify the coefficients a, b, and c for the quadratic equation \( ax^2 + bx + c = 0 \):

a = -1, b = -3, c = -3

Step 4: Use the quadratic formula to solve for x:

\( x = \frac{{-b \pm \sqrt{{b^2 – 4ac}}}}{{2a}} \)

Substitute a = -1, b = -3, and c = -3:

x = (3 ± √((-3)² – 4(-1)(-3)))/(-2)

x = (3 ± √(9 – 12))/(-2)

x = (3 ± √(-3))/(-2)

x = (3 ± i√3)/(-2)

x = -3/2 ± i√3/2

Summary: The solutions to the equation -x² = 3(x + 1) are two complex numbers:

x = -3/2 – i√3/2 and x = -3/2 + i√3/2.