Step 1: Expand using $(a+b)^2=a^2+2ab+b^2$.
$$\left(2+\sqrt{2}\right)^{2}=2^{2}+2(2)(\sqrt{2})+(\sqrt{2})^{2}$$
Step 2: Simplify each term.
$$=4+4\sqrt{2}+2=6+4\sqrt{2}$$
Step 3: Here 6 is rational and $4\sqrt{2}$ is irrational.
Step 4: Rational $+$ irrational $=$ irrational.
Hence $\left(2+\sqrt{2}\right)^{2}=6+4\sqrt{2}$ is an irrational number. Correct option is (B).