If so, the point to get is that
(where m happens to be the repeated root of so that and also
has been shown to be a solution to the differential equation because it has been shown to produce the equation. I.e. what has been shown is that setting
come to zero.
Maybe the 'Now we know' line would be slightly clearer as:
"Now we were assuming that the m in the exponent is equal to which, with the help of , is going to make both of and come to zero.
Or maybe not! But you could also dispense with m in the first place and set,
... as here: Pauls Online Notes : Differential Equations - Repeated Roots
Otherwise, if the issue is just the putting together of this solution with the previous one at the top of the page, notice that (A + B) in the original version of the first solution has been replaced by A alone, and B in the second solution has nothing to do with B at the top of the page.