The Pythagorean theorem applies, so the sum of the squares of the leg lengths is equal to the square of the hypotenuse length.
... 8² + (n+4)² = (n+8)²
... 64 + n² + 8n + 16 = n² + 16n + 64
... 0 = 8n -16 . . . . . . subtract the left side
... 0 = n -2 . . . . . . . . divide by the coefficient of n
... 2 = n . . . . . . . . . . .add 2
The left side has length n+4 = 2+4 = 6.
The hypotenuse has length n+8 = 2+8 = 10.