How do you find $\left( fg \right)\left( 5 \right)+f\left( 4 \right)$ when $f\left( x \right)={{x}^{2}}+1$ and $g\left( x \right)=x-4$?

Hint: In this problem we need to calculate the value of the given expression by using the definition of the given functions. In the expression we have two values, first one is $fg$, second one is $f$. In the problem we have definitions of $f$ and $g$. From these values we will calculate the value of $fg$. After having the value of $fg$ we will use this and calculate the value of the given expression.

Complete step by step answer:
Given function definitions are $f\left( x \right)={{x}^{2}}+1$, $g\left( x \right)=x-4$.
Now the value of the function $fg$ will be calculated by multiplying the function $f$ with $g$, then we will get
$\Rightarrow fg=\left( {{x}^{2}}+1 \right)\left( x-4 \right)$
Using the distribution law of the multiplication in the above equation, then we will have
$\begin{align}
  & \Rightarrow fg={{x}^{2}}\left( x-4 \right)+1\left( x-4 \right) \\
 & \Rightarrow fg={{x}^{3}}-4{{x}^{2}}+x-4 \\
\end{align}$
Now the value of $fg\left( 5 \right)$ can be calculated by substituting $x=5$ in the above equation, then we will get
$\begin{align}
  & \Rightarrow fg={{5}^{3}}-4\times {{5}^{2}}+5-4 \\
 & \Rightarrow fg=125-4\times 25+1 \\
 & \Rightarrow fg=126-100 \\
 & \Rightarrow fg=26 \\
\end{align}$
Now the value of $f\left( 4 \right)$ can be calculated by substituting $x=4$ in the function value $f\left( x \right)={{x}^{2}}+1$, then we will have
$\begin{align}
  & \Rightarrow f\left( 4 \right)={{4}^{2}}+1 \\
 & \Rightarrow f\left( 4 \right)=16+1 \\
 & \Rightarrow f\left( 4 \right)=17 \\
\end{align}$
In this problem they have asked to calculate the value of $\left( fg \right)\left( 5 \right)+f\left( 4 \right)$. So, the value of $\left( fg \right)\left( 5 \right)+f\left( 4 \right)$ will be
$\begin{align}
  & \Rightarrow \left( fg \right)\left( 5 \right)+f\left( 4 \right)=26+17 \\
 & \Rightarrow \left( fg \right)\left( 5 \right)+f\left( 4 \right)=43 \\
\end{align}$

Note: We can also solve this problem in another method that is without calculating the value of $fg$. We can directly solve the problem by following the below procedure.
$\begin{align}
  & \Rightarrow \left( fg \right)\left( 5 \right)+f\left( 4 \right)=\left( {{5}^{2}}+1 \right)\left( 5-4 \right)+\left( {{4}^{2}}+1 \right) \\
 & \Rightarrow \left( fg \right)\left( 5 \right)+f\left( 4 \right)=\left( 25+1 \right)\left( 1 \right)+\left( 16+1 \right) \\
 & \Rightarrow \left( fg \right)\left( 5 \right)+f\left( 4 \right)=26+17 \\
 & \Rightarrow \left( fg \right)\left( 5 \right)+f\left( 4 \right)=43 \\
\end{align}$
From both the methods we got the same result.