Solve $ {\left( {a + b + c} \right)^3} $

Hint: Here given an expression, and we have to find the cube of the given expression. Finding the cube of an expression is nothing but we have to multiply the given expression with itself for three times. That is given an expression, the cube of the expression is the product of the given expression and the square of the expression.
The formula used here is :
 $ \Rightarrow {\left( {a + b + c} \right)^2} = $ \[ \left( {{a^2} + ab + ac + ab + {b^2} + bc + ac + bc + {c^2}} \right)\]

Complete step-by-step answer:
Given an expression, which is $ {\left( {a + b + c} \right)^3} $ .
We are asked to solve the above expression.
Consider the given expression as given below:
 $ \Rightarrow {\left( {a + b + c} \right)^3} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ac $
Expanding the above expression, as given below
 $ \Rightarrow {\left( {a + b + c} \right)^2}\left( {a + b + c} \right) $
Substituting the formula of $ {\left( {a + b + c} \right)^2} $ , in the above expression, as given below:
 \[ \Rightarrow \left( {{a^2} + ab + ac + ab + {b^2} + bc + ac + bc + {c^2}} \right)\left( {a + b + c} \right)\]
 $ \Rightarrow \left( {{a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ac} \right)\left( {a + b + c} \right) $
Now expanding and simplifying the terms inside the brackets, as given below:
 $ \Rightarrow {a^2}\left( {a + b + c} \right) + {b^2}\left( {a + b + c} \right) + {c^2}\left( {a + b + c} \right) + 2ab\left( {a + b + c} \right) + 2bc\left( {a + b + c} \right) + 2ac\left( {a + b + c} \right) $
 $ \Rightarrow {a^3} + {a^2}b + {a^2}c + a{b^2} + {b^3} + {b^2}c + a{c^2} + b{c^2} + {c^3} + 2{a^2}b + 2a{b^2} + 2ac + 2abc + 2{b^2}c + 2b{c^2} + 2{a^2}c + 2abc + 2a{c^2} $
Grouping the like terms and unlike terms together, in order to simplify the expression, as given below:
 $ \Rightarrow {a^3} + {b^3} + {c^3} + 3{a^2}b + 3{a^2}c + 3a{b^2} + 3{b^2}c + 3a{c^2} + 3b{c^2} + 6abc $
 $ \Rightarrow {a^3} + {b^3} + {c^3} + 3{a^2}b + 3a{b^2} + 3{b^2}c + 3b{c^2} + 3a{c^2} + 3{a^2}c + 6abc $
Here when considered the two terms $ 3{a^2}b + 3a{b^2} $ , here taking the terms $ 3ab $ common which gives the simplified expression of $ 3{a^2}b + 3a{b^2} = 3ab\left( {a + b} \right) $ .
Similarly for $ 3{b^2}c + 3b{c^2} = 3bc\left( {b + c} \right) $ ,
Also for the expression $ 3a{c^2} + 3{a^2}c = 3ac\left( {a + c} \right) $ .
Now substitute the simplified expressions obtained, as given below:
 \[ \Rightarrow {a^3} + {b^3} + {c^3} + 3ab\left( {a + b} \right) + 3ac\left( {a + c} \right) + 3bc\left( {b + c} \right) + 6abc\]
 $ \therefore {\left( {a + b + c} \right)^3} = {a^3} + {b^3} + {c^3} + 3ab\left( {a + b} \right) + 3ac\left( {a + c} \right) + 3bc\left( {b + c} \right) + 6abc $

Final Answer: The expansion of $ {\left( {a + b + c} \right)^3} = {a^3} + {b^3} + {c^3} + 3ab\left( {a + b} \right) + 3ac\left( {a + c} \right) + 3bc\left( {b + c} \right) + 6abc $

Note:
Please note that the expansion of $ {\left( {a + b + c} \right)^3} $ can also be one in another way but with a slight change. That is instead of expanding the given expression $ {\left( {a + b + c} \right)^3} $ as the product of $ {\left( {a + b + c} \right)^2} $ and $ \left( {a + b + c} \right) $ . This can also be solved by expanding the expression $ {\left( {a + b + c} \right)^3} $ as the product of $ \left( {a + b + c} \right) $ for 3 times as : $ \left( {a + b + c} \right)\left( {a + b + c} \right)\left( {a + b + c} \right) $ which can be written as given below:
 $ \Rightarrow {\left( {a + b + c} \right)^3} = \left( {a + b + c} \right)\left( {a + b + c} \right)\left( {a + b + c} \right) $ , expanding and solving this expression also gives the same final answer.