... so, down to but not including, "giving as a solution" ...?

If so, the point to get is that

(wherem 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

(where... )

makes

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with the previous one at the top of the page, notice that (A + B) in the original version of the first solution has been replaced by A alone, and B in the second solution has nothing to do with B at the top of the page.