To find the set of values for $k$ such that the line $y = x + k$ is tangent to the parabola $y = x^2 + x - 2$, we need to follow these steps:

1. Set the equations equal to each other (since at the point of tangency, the $y$-coordinates will be equal):

$x + k = x^2 + x - 2$

2. Rearrange the equation to form a quadratic equation:

$0 = x^2 + x - 2 - (x + k)$

Simplifying:

$0 = x^2 + x - 2 - x - k$

$0 = x^2 - 2 - k$

$x^2 = k + 2$

Thus, the equation simplifies to:

[ x^2 = k + 2. This