How do I do this A level Maths q?

6 + x ≥ x²   and   x + 2 < x²
x² ≤ 6 + x   and   x² > x + 2
x² - x - 6 ≤ 6 + x - x - 6   and   x² - x - 2 > x + 2 - x - 2
x² - x - 6 ≤ 0   and   x² - x - 2 > 0
(x + 2)(x - 3) ≤ 0   and   (x + 1)(x - 2) > 0
-2 ≤ x ≤ 3   and   (x < -1 or x > 2)

Refer to the number line below.
Range of x:  -2 ≤ x < -1  or  2 < x ≤ 3

Assuming you mean 6 + x ≥ x^2 and x + 2 < x^2,
6 + x ≥ x^2
0 ≥ x^2 - x - 6
0 ≥ (x - 3)(x + 2)
-2 ≤ x ≤ 3
x ∈ [-2, 3]

x + 2 < x^2
0 < x^2 - x - 2
0 < (x - 2)(x + 1)
x > 2 or x < -1
x ∈ (-∞ , -1) U (2,∞ )

x ∈ [-2, 3] ∩ ( (-∞ , -1) U (2,∞ ) )
x ∈ [-2, -1) U (2, 3]