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