Expand the following using suitable identities: \[{\left( { - 2x + 3y + 2z} \right)^2}\].

Hint: Here, we need to find the square of the sum of the numbers \[ - 2x\], \[3y\], and \[2z\]. We will use the algebraic identity for the square of the sum of three numbers. Then, we will simplify the expression to get the required expansion of \[{\left( { - 2x + 3y + 2z} \right)^2}\]. The square of sum of three numbers \[a\], \[b\], and \[c\] is given by the algebraic identity \[{\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2\left( {ab + bc + ca} \right)\].

Formula Used: The square of sum of three numbers \[a\], \[b\], and \[c\] is given by the algebraic identity \[{\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2\left( {ab + bc + ca} \right)\].

Complete step-by-step answer:
We will use the algebraic identity for the square of the sum of three numbers.
The square of sum of three numbers \[a\], \[b\], and \[c\] is given by the algebraic identity \[{\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2\left( {ab + bc + ca} \right)\].
We need to find the square of the sum of the numbers \[ - 2x\], \[3y\], and \[2z\].
Substituting \[a = - 2x\], \[b = 3y\], and \[c = 2z\] in the algebraic identity, we get
\[ \Rightarrow {\left( { - 2x + 3y + 2z} \right)^2} = {\left( { - 2x} \right)^2} + {\left( {3y} \right)^2} + {\left( {2z} \right)^2} + 2\left[ {\left( { - 2x} \right)\left( {3y} \right) + \left( {3y} \right)\left( {2z} \right) + \left( {2z} \right)\left( { - 2x} \right)} \right]\]
First, we will simplify the expression in the parentheses.
Multiplying the terms in the parentheses, we get
\[ \Rightarrow {\left( { - 2x + 3y + 2z} \right)^2} = {\left( { - 2x} \right)^2} + {\left( {3y} \right)^2} + {\left( {2z} \right)^2} + 2\left[ { - 6xy + 6yz - 4xz} \right]\]
Multiplying 2 by \[ - 6xy + 6yz - 4xz\] using the distributive law of multiplication, we get
\[ \Rightarrow {\left( { - 2x + 3y + 2z} \right)^2} = {\left( { - 2x} \right)^2} + {\left( {3y} \right)^2} + {\left( {2z} \right)^2} - 12xy + 12yz - 8xz\]
Now, we will simplify the other terms of the expression.
Applying the exponents on the bases, we get
\[\therefore {\left( { - 2x + 3y + 2z} \right)^2} = 4{x^2} + 9{y^2} + 4{z^2} - 12xy + 12yz - 8xz\]
Since there are no like terms, we cannot simplify the expression further.
Therefore, the expansion of \[{\left( { - 2x + 3y + 2z} \right)^2}\] is \[4{x^2} + 9{y^2} + 4{z^2} - 12xy + 12yz - 8xz\].

Note: We have used the distributive law of multiplication to find the product of 2 and \[ - 6xy + 6yz - 4xz\]. The distributive law of multiplication states that \[a\left( {b + c + d} \right) = a \cdot b + a \cdot c + a \cdot d\].
We cannot simplify \[4{x^2} + 9{y^2} + 4{z^2} - 12xy + 12yz - 8xz\] 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 \[4{x^2}\] to \[9{y^2}\] or \[ - 12xy\].