Two integers differ by 2 and sum of their squares is 52. The integers are :

1. 4 and 6

2. 4 or 6

3. -4 or 6

4. -4 and -6 or 6 and 4

Let two integers be x and x - 2.

Give,

Sum of squares = 52

⇒ x² + (x - 2)² = 52

⇒ x² + x² + 4 - 4x = 52

⇒ 2x² - 4x + 4 - 52 = 0

⇒ 2x² - 4x - 48 = 0

⇒ 2(x² - 2x - 24) = 0

⇒ x² - 2x - 24 = 0

⇒ x² - 6x + 4x - 24 = 0

⇒ x(x - 6) + 4(x - 6) = 0

⇒ (x + 4)(x - 6) = 0

⇒ x + 4 = 0 or x - 6 = 0

⇒ x = -4 or x = 6.

If x = -4,

x - 2 = -4 - 2 = -6.

If x = 6,

x - 2 = 6 - 2 = 4.

Numbers = -4 and -6 or 6 and 4.

Hence, Option 4 is the correct option.