How do you expand \[{{\left( b+2 \right)}^{2}}\] ?

Hint: From the given question we have to expand the binomial\[{{\left( b+2 \right)}^{2}}\]. To expand this, we have to use binomial theorem i.e., the expansion of \[{{\left( a+b \right)}^{n}}=\sum\limits_{k=0}^{n}{{}^{n}{{C}_{k}}.\left( {{a}^{n-k}}{{b}^{k}} \right)}\]. Here we have to substitute b in place of a and \[2\] in place of b. by this we can expand the above binomial \[{{\left( b+2 \right)}^{2}}\].

Complete step by step solution:
From the given question we have to expand the binomial \[{{\left( b+2 \right)}^{2}}\]
As we know that we have to expand this by using binomial theorem. Binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial \[{{\left( a+b \right)}^{n}}\] into a sum involving terms of the form \[c{{a}^{x}}{{b}^{y}}\], where the exponents x and y are nonnegative integers with \[x+y=n\], and the coefficient c of each term is a specific positive integer depending on n and x. the coefficient c in the term of \[c{{a}^{x}}{{b}^{y}}\] is known as the binomial coefficient.
Now, by using binomial theorem we have to expand the binomial \[{{\left( b+2 \right)}^{2}}\].
\[\Rightarrow {{\left( b+2 \right)}^{2}}=\sum\limits_{k=0}^{2}{\dfrac{2!}{\left( 2-k \right)!k!}.\left( {{b}^{2-k}} \right)}.{{\left( 2 \right)}^{k}}\]
Now we have to expand the summation.
\[\Rightarrow {{\left( b+2 \right)}^{2}}=\dfrac{2!}{\left( 2-0 \right)!0!}.\left( {{b}^{2-0}} \right){{.2}^{0}}+\dfrac{2!}{\left( 2-1 \right)!1!}.\left( {{b}^{2-1}} \right).{{\left( 2 \right)}^{1}}+\dfrac{2!}{\left( 2-2 \right)!2!}.\left( {{b}^{2-2}} \right).{{\left( 2 \right)}^{2}}\]
Now, we have to simplify the above form.
\[\Rightarrow {{\left( b+2 \right)}^{2}}=\left( 1.{{\left( 2 \right)}^{0}}.{{b}^{2}} \right)+\left( 2.{{\left( 2 \right)}^{1}}.{{b}^{1}} \right)+\left( 1.{{\left( 2 \right)}^{2}}.{{b}^{0}} \right)\]
After the simplification the above binomial expression is
\[\Rightarrow {{\left( b+2 \right)}^{2}}={{b}^{2}}+4b+4\]
Therefore, this is the required binomial expansion for the given binomial \[{{\left( b+2 \right)}^{2}}\].

Note: Students should know the expansions and binomial theorem. Student should be careful with signs and calculation. Student can do this problem by simply formula as we know that \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]. By substituting in this formula also we can expand the \[{{\left( b+2 \right)}^{2}}\].