What is the value of \[{(a + b)^3} - {(a - b)^3}\] ?
(a). \[{b^3} + 3{a^2}b\]
(b). \[2({b^3} + 3{a^2}b)\]
(c). \[2({a^3} + 3a{b^2})\]
(d). 0

Hint: Recall the formula for \[{(a + b)^3}\] and \[{(a - b)^3}\], which is \[{(a + b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3}\] , and \[{(a - b)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3}\] respectively. Use these formulae to evaluate \[{(a + b)^3} - {(a - b)^3}\].

Complete Step-by-Step solution:
In this problem, we need to evaluate \[{(a + b)^3} - {(a - b)^3}\]. For that first, we find the expansion of \[{(a + b)^3}\] and \[{(a - b)^3}\] and then evaluate them.
First, let us evaluate \[{(a + b)^3}\]. We know that \[{(a + b)^3}\] can be written as \[(a + b)(a + b)(a + b)\]. Hence, we have:
$\Rightarrow$ \[{(a + b)^3} = (a + b)(a + b)(a + b)\]
We multiply the first two terms which is nothing but the expression for \[{(a + b)^2}\].
$\Rightarrow$ \[{(a + b)^3} = (a.a + a.b + b.a + b.b)(a + b)\]
Simplifying the above equation, we get as follows:
$\Rightarrow$ \[{(a + b)^3} = ({a^2} + 2ab + {b^2})(a + b)\]
Now, we multiply the remaining terms to get the final expression. Hence, we have:
$\Rightarrow$ \[{(a + b)^3} = {a^2}.a + 2ab.a + {b^2}.a + {a^2}.b + 2ab.b + {b^2}.b\]
Simplifying the above equation, we get:
$\Rightarrow$ \[{(a + b)^3} = {a^3} + 2{a^2}b + a{b^2} + {a^2}b + 2a{b^2} + {b^3}\]
Adding the common terms together, we have:
$\Rightarrow$ \[{(a + b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3}.............(1)\]
Now, let us evaluate \[{(a - b)^3}\]. We know that \[{(a - b)^3}\] can be written as \[(a - b)(a - b)(a - b)\]. Hence, we have:
$\Rightarrow$ \[{(a - b)^3} = (a - b)(a - b)(a - b)\]
We multiply the first two terms which is nothing but the expression for \[{(a - b)^2}\].
$\Rightarrow$ \[{(a - b)^3} = (a.a - a.b - b.a + b.b)(a - b)\]
Simplifying the above equation, we get as follows:
$\Rightarrow$ \[{(a - b)^3} = ({a^2} - 2ab + {b^2})(a - b)\]
Now, we multiply the remaining terms to get the final expression. Hence, we have:
$\Rightarrow$ \[{(a - b)^3} = {a^2}.a - 2ab.a + {b^2}.a - {a^2}.b + 2ab.b - {b^2}.b\]
Simplifying the above equation, we get:
$\Rightarrow$ \[{(a - b)^3} = {a^3} - 2{a^2}b + a{b^2} - {a^2}b + 2a{b^2} - {b^3}\]
Adding the common terms together, we have:
$\Rightarrow$ \[{(a - b)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3}.............(2)\]
Now, we subtract equation (2) from equation (1) to obtain the value of \[{(a + b)^3} - {(a - b)^3}\]. Hence, we have as follows:
$\Rightarrow$ \[{(a + b)^3} - {(a - b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} - ({a^3} - 3{a^2}b + 3a{b^2} - {b^3})\]
Taking the minus sign inside the bracket, we get:
$\Rightarrow$ \[{(a + b)^3} - {(a - b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} - {a^3} + 3{a^2}b - 3a{b^2} + {b^3}\]
Canceling the common terms, we have:
$\Rightarrow$ \[{(a + b)^3} - {(a - b)^3} = 3a{b^2} + {b^3} + 3{a^2}b + {b^3}\]
Simplifying, we have:
$\Rightarrow$ \[{(a + b)^3} - {(a - b)^3} = 2{b^3} + 6{a^2}b\]
Taking 2 as a common term outside, we have:
$\Rightarrow$ \[{(a + b)^3} - {(a - b)^3} = 2({b^3} + 3{a^2}b)\]
Hence, option (b) is the correct answer.

Note: Note that instead of subtracting the two terms, if you add, then you will get the result as in option (c), which is wrong. Hence, evaluate carefully.