Expand \[{\left( {x + 1} \right)^3}\] using an identity.

Hint: Here, we need to expand the expression \[{\left( {x + 1} \right)^3}\]. We will use the algebraic identity for the cube of the sum of two numbers. Then, we will simplify the expression to get the required expansion of \[{\left( {x + 1} \right)^3}\].

Formula Used: The cube of the sum of two numbers \[a\] and \[b\] is given by the algebraic identity \[{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)\].

Complete step-by-step answer:
We need to find the cube of the sum of the numbers \[x\] and 1.
Substituting \[a = x\] and \[b = 1\] in the algebraic identity \[{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)\], we get
\[ \Rightarrow {\left( {x + 1} \right)^3} = {\left( x \right)^3} + {\left( 1 \right)^3} + 3\left( x \right)\left( 1 \right)\left( {x + 1} \right)\]
First, we will simplify the third expression on the right hand side.
Multiplying the terms, we get
\[ \Rightarrow {\left( {x + 1} \right)^3} = {\left( x \right)^3} + {\left( 1 \right)^3} + 3x\left( {x + 1} \right)\]
Multiplying \[3x\] by \[x + 1\] using the distributive law of multiplication, we get
\[ \Rightarrow {\left( {x + 1} \right)^3} = {\left( x \right)^3} + {\left( 1 \right)^3} + 3{x^2} + 3x\]
Now, we will simplify the other terms of the expression.
Applying the exponents on the bases, we get
\[ \Rightarrow {\left( {x + 1} \right)^3} = {x^3} + 1 + 3{x^2} + 3x\]
Rewriting the expression, we get
\[ \Rightarrow {\left( {x + 1} \right)^3} = {x^3} + 3{x^2} + 3x + 1\]
Since there are no like terms, we cannot simplify the expression further.
\[\therefore \] The expansion of \[{\left( {x + 1} \right)^3}\] is \[{x^3} + 3{x^2} + 3x + 1\].

Note: We have used the distributive law of multiplication to find the product of \[3x\] and \[x + 1\]. The distributive law of multiplication states that \[a\left( {b + c} \right) = a \cdot b + a \cdot c\].
We cannot simplify \[{x^3} + 3{x^2} + 3x + 1\] further because there are no like terms in the expression. Like terms are the terms whose variables and their exponents are the same. For example, \[100x,150x,240x,600x\] all have the variable \[x\] raised to the exponent 1. Terms which are not like, and have different variables, or different degrees of variables cannot be added together. For example, it is not possible to add \[{x^3}\] to \[3{x^2}\] or \[3x\].