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b\left(2c+1\right)+d\left(2c+1\right)
Do the grouping 2bc+b+2cd+d=\left(2bc+b\right)+\left(2cd+d\right), and factor out b in the first and d in the second group.
\left(2c+1\right)\left(b+d\right)
Factor out common term 2c+1 by using distributive property.