To solve this inequality, we need to find the values of x that satisfy the inequality:

√2 + x - x^2 > -1

First, let's simplify the inequality:

√2 + x - x^2 > -1
√2 + x - x^2 + 1 > 0
√2 + x - x^2 + 1 + x^2 > x^2
√2 + x + 1 > x^2

Now, let's square both sides of the inequality to get rid of the square root:

(√2 + x + 1)^2 > x^2

Expanding the left side:

(√2 + x + 1)(√2 + x + 1) > x^2
2 + 2√2x + 2x + x^2 + 2√2 + x + √2 > x^2
2 + 2√2x + 2x + x^2 + 2√2 + x + √2 > x^2

Simplify the inequality:

2 + 2√2x + 2x + x^2 + 2√2 + x + √2 > x^2
x^2 + 3x + 2√2 + √2 - 2 > 0
x^2 + 3x + 3√2 - 2 > 0

Now, we need to find the values of x that make this quadratic inequality true. We can solve this inequality by factoring or using the quadratic formula:

x^2 + 3x + 3√2 - 2 > 0

The solutions of this inequality will be the values of x that satisfy the original inequality √2 + x - x^2 > -1.