Answer
The required solution is $48$
Work Step by Step
Find $f\left( 5 \right)$.
Since $5>2$ ,
$f\left( x \right)={{x}^{2}}+2x-1$
Put $5$ for $x$ in $f\left( x \right)$:
$\begin{align}
& f\left( 5 \right)={{\left( 5 \right)}^{2}}+2\left( 5 \right)-1 \\
& =25+10-1 \\
& =34
\end{align}$
Find $f\left( -5 \right)$.
Since $-5<2$ ,
$f\left( x \right)=3x+1$
Put $-5$ for $x$ in $f\left( x \right)$:
$\begin{align}
& f\left( -5 \right)=3\left( -5 \right)+1 \\
& =-15+1 \\
& =-14
\end{align}$
Hence,
$\begin{align}
& f\left( 5 \right)-f\left( -5 \right)=34-\left( -14 \right) \\
& =48
\end{align}$
