Actually, your work and answer are almost correct, but there was a small mistake in your first equation. The correct equation should be:
n^2 + (n+1)^2 = 85
Let's go through the solution step by step:
1. Write down the equation: n^2 + (n+1)^2 = 85
2. Expand the equation: n^2 + (n+1)(n+1) = 85
n^2 + (n^2 + 2n + 1) = 85
2n^2 + 2n + 1 = 85
3. Rearrange the equation: 2n^2 + 2n - 84 = 0
4. Simplify the equation by dividing all terms by 2: n^2 + n - 42 = 0
5. Factor the equation: (n - 6)(n + 7) = 0
6. Set each factor equal to zero: n - 6 = 0 or n + 7 = 0
n = 6 or n = -7
Since we are looking for positive integers, we can disregard the solution n = -7.
Therefore, the consecutive positive integers that satisfy the equation are n = 6 and n+1 = 7.