Step 1: We can simplify equation by multiplying both sides by -1. After multiplying we have the following equation:
$$ \begin{aligned} -x^2-x-1 &= 0 \,\,\, / \color{orangered}{\cdot \, -1 } \\[0.9 em ] x^2+x+1 &=0 \end{aligned} $$Step 2: Read the values of $ a $, $ b $, and $ c $ from the quadratic equation: $ a $ is the number in front of $ x^2 $, $ b $ is the number in front of $ x $, $ c $ is the number at the end. In our case:
$$ a = 1, \,\, b = 1, \,\, c = 1 $$Step 3: Plug in the values for $ a $, $ b $, and $ c $ into the quadratic formula.
$$ \begin{aligned} x_1,x_2 &= \frac{-b \pm \sqrt{b^2-4ac}}{2a} \\[1 em] x_1,x_2 &= \frac{ -1 \pm \sqrt{ 1 ^2 - 4 \cdot 1 \cdot 1} }{ 2 \cdot 1 } \end{aligned} $$Step 4: Simplify expression under the square root.
$$ x_1,x_2 = \frac{ -1 \pm \sqrt{ -3 } }{ 2 } $$Step 5: Solve for $ x $
$$ \begin{aligned} & \color{blue}{ x_2 = \frac{ -1~+~\sqrt{ -3 } }{ 2 } = -\frac{ 1 }{ 2 }+\frac{\sqrt{ 3 }}{ 2 }i } \\\\ & \color{blue}{ x_1 = \frac{ -1~-~\sqrt{ -3 } }{ 2 } = -\frac{ 1 }{ 2 }-\frac{\sqrt{ 3 }}{ 2 }i } \end{aligned} $$