Answer
The standard form of the expression ${{\left( 2+i \right)}^{2}}-{{\left( 3-i \right)}^{2}}$ is $-5+10i$.
Work Step by Step
Consider the expression,${{\left( 2+i \right)}^{2}}-{{\left( 3-i \right)}^{2}}$
Use the formula ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ and ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$.
\[\begin{align}
& {{\left( 2+i \right)}^{2}}-{{\left( 3-i \right)}^{2}}=\left( {{2}^{2}}+2\cdot 2\cdot i+{{i}^{2}} \right)-\left( {{3}^{2}}-2\cdot 3\cdot i+{{i}^{2}} \right) \\
& =\left( 4+4i+{{i}^{2}} \right)-\left( 9-6i+{{i}^{2}} \right) \\
& =4+4i+{{i}^{2}}-9+6i-{{i}^{2}}
\end{align}\]
Combine real and imaginary terms.
\[\begin{align}
& {{\left( 2+i \right)}^{2}}-{{\left( 3-i \right)}^{2}}=4-9+4i+6i+{{i}^{2}}-{{i}^{2}} \\
& =-5+10i
\end{align}\]
Therefore, the standard form of the expression ${{\left( 2+i \right)}^{2}}-{{\left( 3-i \right)}^{2}}$ is $-5+10i$.
