Question

a whole number increased by its square is two more than twice itself. find the number

Answer 1

Let x be the number.
=> x + x² = 2x + 2
=> x² - x - 2 = 0
=> (x + 1)(x - 2) = 0
---> x = -1, or 2
You didn't mention negatives, but guessing you'd want "2".