How To Expand (a+b+c)(d+e+f)(g+h+i)

Firstly you have to know what (a+b+c)(d+e+f) is. You can expand this expression using a rectangle:

(a+b+c)(d+e+f)

So you know that: \left( a+b+c \right) \left( d+e+f \right) =ad+ae+af+bd+be+bf+cd+ce+cf

Next you’d have to multiply (a+b+c)(d+e+f) by (g+h+i) using another rectangle:

(a+b+c)(d+e+f)(g+h+i)

And from here you’d figure out that:

\left( a+b+c \right) \left( d+e+f \right) \left( g+h+i \right) \\ \\ =\left[ \left( a+b+c \right) \left( d+e+f \right)  \right] \left( g+h+i \right) \\ \\ =\left[ ad+ae+af+bd+be+bf+cd+ce+cf \right] \left( g+h+i \right) \\ \\ =adg+aeg+afg+bdg+beg+bfg+cdg+ceg+cfg\\ \\ +adh+aeh+afh+bdh+beh+bfh+cdh+ceh+cfh\\ \\ +adi+aei+afi+bdi+bei+bfi+cdi+cei+cfi\\ \\ =ag\left( d+e+f \right) +bg\left( d+e+f \right) +cg\left( d+e+f \right) \\ \\ +ah\left( d+e+f \right) +bh\left( d+e+f \right) +ch\left( d+e+f \right) \\ \\ +ai\left( d+e+f \right) +bi\left( d+e+f \right) +ci\left( d+e+f \right) \\ \\ =\left( d+e+f \right) \left[ ag+bg+cg+ah+bh+ch+ai+bi+ci \right] \\ \\ =\left( d+e+f \right) \left[ g\left( a+b+c \right) +h\left( a+b+c \right) +i\left( a+b+c \right)  \right] \\ \\ =\left( d+e+f \right) \left[ \left( a+b+c \right) \left( g+h+i \right)  \right] \\ \\ =\left( a+b+c \right) \left( d+e+f \right) \left( g+h+i \right)

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