To solve the equation \( x^2 - 20x + 90 = -x \), we first bring all terms to one side of the equation to set it to zero.
Starting with the original equation:
\[ x^2 - 20x + 90 + x = 0 \]
This simplifies to:
\[ x^2 - 19x + 90 = 0 \]
Next, we need to factor the quadratic equation \( x^2 - 19x + 90 \). We are looking for two numbers that multiply to \( 90 \) (the constant term) and add to \( -19 \) (the coefficient of \( x \)). The numbers \( -9 \) and \( -10 \) meet these criteria:
\[ -9 \cdot -10 = 90 \quad \text{and} \quad -9 + -10 = -19 \]
So, we can factor the quadratic as follows:
\[ (x - 9)(x - 10) = 0 \]
Setting each factor equal to zero gives us the possible solutions for \( x \):
- \( x - 9 = 0 \) → \( x = 9 \)
- \( x - 10 = 0 \) → \( x = 10 \)
Therefore, the solutions are:
\[ \boxed{9 \text{ and } 10} \]