Answer:smaller integer = 7The larger integer = 9Step-by-step explanation:* Lets explain how to solve the problem- There are two integers- Assume that the smaller is x and the larger is y- The difference between them is 2∵ The y is the larger and x is the smaller∵ y - x = 2 ⇒ (1)- If the smaller is added to the square of the larger, the sum is 88∵ The square of the larger is y²∴ x + y² = 88 ⇒ (2)* Lets use the substitute method to solve the problem- Use equation (1) to find x in terms of y∵ y - x = 2 ⇒ add x to 2 sides and subtract 2 from both sides∴ y - 2 = x ⇒ switch the two sides∴ x = y - 2 ⇒ (3)- Substitute x in equation (2) by equation (3)∴ (y - 2) + y² = 88 ⇒ subtract 88 from both sides∴ y - 2 + y² - 88 = 0 ⇒ add the like terms and arrange the terms∴ y² + y - 90 = 0* Lets factorize it∵ y² = y × y∵ 90 = 9 × 10∵ y × 9 = 9y and y × 10 = 10y∵ 10y - 9y = y∴ y² + y - 90 = (y + 10)(y - 9) ∵ y² + y - 90 = 0∴ (y + 10)(y - 9) = 0- Equate each bracket by 0∴ y + 10 = 0 and y - 9 = 0∵ y + 10 = 0 ⇒ subtract 10 from both sides∴ y = -10∵ y - 9 = 0 ⇒ add 9 to both sides∴ y = 9∵ The numbers are positive we will rejected y = -10∴ y = 9- Substitute the values of y in equation (3) to find x∵ x = y - 2∴ x = 9 - 2 = 7∴ x = 7* The smaller integer = 7* The larger integer = 9