The answer is B+C but When I tried to solve I got a different answer, which is B.
The way I solved
AB+B((B+C')+B'C)
=AB+B(B+C')+BB'C
=AB+BB+BC'+BB'C
=(BB'C=0,BB=B) then
AB+B+BC'
Taking B out frm all terms
B(A+1+C')
=B
[A+1+C'=1]
What is the problem in my solution
Your solution looks all good. That means textbook answer $B+C$ must be incorrect, or there is a typo in the given expression.
Let's check this by plugging in $B=0$ and $C=1$:
$B+C = 0+1 = 1$
$AB+B((B+C')+B'C) = A0 + 0(...) = 0+0 = 0$
Thus $B+C$ cannot be the simplified form. However, you would get the textbook answer if the given expression had been: $$AB+B(B+C')+B'C $$