The (unsigned) vertical distance between the two curves is
|(3 - x²) - (x + 1)| = |x² + x - 2|
Since
x² + x - 2 = (x + 2) (x - 1)
we know the distances between the curves at the endpoints of the interval [-2, 1] are both zero.
For all other x in (-2, 1), the quadratic x² + x - 2 is negative. (Consider x = 0, for instance, which gives a value of -2 < 0.) So by definition of absolute value, we have
|x² + x - 2| = - (x² + x - 2) = 2 - x - x²
Complete the square:
2 - x - x² = 2 + 1/4 - (1/4 + x + x²)
2 - x - x² = 9/4 - (1/2 + x)²
Then the maximum distance between the curves is 9/4, which occurs at x = -1/2.
