Factorise the expression: ${\left( {a - b} \right)^2} + 4ab$

Hint: In the given question, we have to factorise the given algebraic expression. First, we expand the whole square term using the algebraic identity ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$. Then, we condense the entire algebraic expression using the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ so as to factorize the expression.

Complete step-by-step solution:
Given question requires us to factorize the expression: ${\left( {a - b} \right)^2} + 4ab$.
So, let the expression ${\left( {a - b} \right)^2} + 4ab$ be $P$.
We will start expanding the whole square term using the algebraic identity ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$. So, we get,
$ \Rightarrow {\left( {a - b} \right)^2} + 4ab = \left( {{a^2} - 2ab + {b^2}} \right) + 4ab$
Now, we add up the like terms and simplify the algebraic expression. Hence, we get,
$ \Rightarrow {\left( {a - b} \right)^2} + 4ab = {a^2} + 2ab + {b^2}$
Now, we can factorise the given polynomial using the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$. So, we condense the entire algebraic expression using the identity to get,
$ \Rightarrow {\left( {a - b} \right)^2} + 4ab = {\left( {a + b} \right)^2}$
Now, we know that the square of a term is obtained by multiplying that particular term with itself. So, we get,
$ \Rightarrow {\left( {a - b} \right)^2} + 4ab = \left( {a + b} \right)\left( {a + b} \right)$
So, the factored form of the given expression $P = {\left( {a - b} \right)^2} + 4ab$ is $\left( {a + b} \right)\left( {a + b} \right)$.
Hence, the option (C) is the correct answer.

Note: Similar to quadratic polynomials, quadratic expressions can also be solved using the factorisation method. We must know the applications of the algebraic identities so as to solve such problems. One must take care of the calculations while taking common terms out of the brackets and factoring the expression so as to be sure of the answer. We must know the simplification rules to obtain the final answer.