When the sum of a number plus 3 is squared it is 11 more than the sum of the number plus 2 when squared.?

Let the number be ( x ). The equation can be set up as follows: ( (x + 3)^2 = (x + 2)^2 + 11 ). Expanding both sides gives ( x^2 + 6x + 9 = x^2 + 4x + 4 + 11 ). Simplifying leads to ( 6x + 9 = 4x + 15 ), which further simplifies to ( 2x = 6 ), resulting in ( x = 3 ).