Formula use: \left(a+b\right)^3=a^3+3.a^2.b+3.a.b^2+b^3
Formula use: \left(a+b\right)^2=a^2+2.a.b+b^2

Explanation

Here, given expression is : a+2b+c
Now, we find the cubes with the help of formula :
\{\left(a\right)+\left(2b+c\right)\}^3
\rightarrow\left(a\right)^3+3.\left(a\right)^2.\left(2b+c\right)+3.a.\left(2b+c\right)^2+\left(2b+c\right)^3 _{[use formula]}
\rightarrow a^3+3a^2.\left(2b+c\right)+3a.\{\left(2b\right)^2+2.2b.c+\left(c\right)^2\}+\{\left(2b\right)^3+3.\left(2b\right)^2.c+3.2b.\left(c\right)^2+\left(c\right)^3\}
\rightarrow a^3+6a^2b+3a^2c+3a.\left({4b}^2+4bc+c^2\right)+\left(8b^3+12b^2c+6bc^2+c^3\right)
\rightarrow a^3+6a^2b+3a^2c+12ab^2+12abc+3ac^2+8b^3+12b^2c+6bc^2+c^3
Ans: a^3+6a^2b+3a^2c+12ab^2+12abc+3ac^2+8b^3+12b^2c+6bc^2+c^3 .