Evaluate the given difference: $ {{\left( a+b \right)}^{3}}-{{\left( a-b \right)}^{3}} $ ?
(a) $ 2a\left( 3{{a}^{2}}+{{b}^{2}} \right) $
(b) $ 2b\left( 3{{a}^{2}}+{{b}^{2}} \right) $
(c) $ 2\left( {{a}^{3}}+{{b}^{3}} \right) $
(d) None of these

Hint: We start solving the problem by assigning variables to the given difference and the terms present in it. We then expand $ {{\left( a+b \right)}^{3}} $ and $ {{\left( a-b \right)}^{3}} $ by following the standard binomial expansion. We then subtract the obtained expansions of $ {{\left( a+b \right)}^{3}} $ and $ {{\left( a-b \right)}^{3}} $ . We then take the common terms from the obtained result to get the required answer.

Complete step by step answer:
According to the problem, we need to find the result of the given difference: $ {{\left( a+b \right)}^{3}}-{{\left( a-b \right)}^{3}} $ .
Let us assume the given difference as $ d=p-q $ , where $ p={{\left( a+b \right)}^{3}} $ and $ q={{\left( a-b \right)}^{3}} $ ---(1).
Let us first solve for p and q to get the result for d.
We have $ p={{\left( a+b \right)}^{3}} $ .
 $ \Rightarrow p={{\left( a \right)}^{3}}+3{{\left( a \right)}^{2}}\left( b \right)+3\left( a \right){{\left( b \right)}^{2}}+{{\left( b \right)}^{3}} $ .
 $ \Rightarrow p={{a}^{3}}+3{{a}^{2}}b+3a{{b}^{2}}+{{b}^{3}} $ ---(2).
Now, we have $ q={{\left( a-b \right)}^{3}} $ .
 $ \Rightarrow q={{\left( a \right)}^{3}}+3{{\left( a \right)}^{2}}\left( -b \right)+3\left( a \right){{\left( -b \right)}^{2}}+{{\left( -b \right)}^{3}} $ .
\[\Rightarrow q={{a}^{3}}-3{{a}^{2}}b+3a{{b}^{2}}-{{b}^{3}}\] ---(3).
Let us substitute equations (2) and (3) in equation (1).
So, we get $ d={{a}^{3}}+3{{a}^{2}}b+3a{{b}^{2}}+{{b}^{3}}-\left( {{a}^{3}}-3{{a}^{2}}b+3a{{b}^{2}}-{{b}^{3}} \right) $ .
 $ \Rightarrow d={{a}^{3}}+3{{a}^{2}}b+3a{{b}^{2}}+{{b}^{3}}-{{a}^{3}}+3{{a}^{2}}b-3a{{b}^{2}}+{{b}^{3}} $ .
 $ \Rightarrow d=6{{a}^{2}}b+2{{b}^{3}} $ .
Let us take the common term present in both terms involved in addition.
 $ \Rightarrow d=\left( 2b\times 3{{a}^{2}} \right)+\left( 2b\times {{b}^{2}} \right) $ .
 $ \Rightarrow d=2b\left( 3{{a}^{2}}+{{b}^{2}} \right) $ .
So, we have found the result of difference $ {{\left( a+b \right)}^{3}}-{{\left( a-b \right)}^{3}} $ as $ 2b\left( 3{{a}^{2}}+{{b}^{2}} \right) $ .
 $\therefore$ The correct option for the problem is (b).

Note:
 We should not confuse while performing calculations involving the multiplication of signs as this will make a big difference in the final result. We should perform each step with good precision in order to avoid confusion and calculation mistakes. We should not stop solving the problem after finding the result as $ 6{{a}^{2}}b+2{{b}^{3}} $ which is the most common mistake done by students. Similarly, we can expect problems to find the result of $ {{\left( a+b \right)}^{3}}+{{\left( a-b \right)}^{3}} $ .