Question 251985
Let the two numbers be x and x+1
Their sum is 2x+1
Their difference is 1

You are told
{{{(sum*difference)+8 = x^2 + (x+1)^2}}}
{{{((2x+1)*1) + 8 = x^2 + x^2 + 2x + 1}}}
{{{2x+9 = 2x^2 + 2x + 1}}}
{{{0 = 2x^2 -8}}}
{{{0 = x^2 -4}}}
{{{0 = (x+2)(x-2)}}}
so x = either -2 or 2.
You are told the number is positive, so x must be 2 and x+1 = 3

Check your answer. Is 13 = 4+9?