How do you FOIL $\left( x+3 \right)\left( x+5 \right)$ ?

Hint: In the problem they have mentioned to use the FOIL rule. Before going to the problem, we will first know about the FOIL rule. FOIL rule is used to do multiplication of two expressions which are in two brackets. FOIL is the short form of First, Outside, Inside, Last. The names itself say that we will first multiply the first terms and then outside terms, after that we will multiply inside terms and finally, we will multiply last terms. To obtain the required result we need to sum up all the results we have got in the above process. So, in the given equation we will multiply the terms according to the FOIL rule and get the result.

Complete step by step answer:
Given that,$\left( x+3 \right)\left( x+5 \right)$
Following the FOIL method in the given equation.
Considering the first terms in the given equation, which are $x$, $x$. Multiplying the first terms then we will get
$x.x={{x}^{2}}...\left( \text{i} \right)$
Considering the outer terms in the given equation, which are $x$, $5$. Multiplying the outer terms then we will get
$x.5=5x....\left( \text{ii} \right)$
Considering the inner terms in the given equation, which are $x$, $3$. Multiplying the inner terms then we will get
$3.x=3x...\left( \text{iii} \right)$
Considering the last terms in the given equation, which are $5$, $3$. Multiplying the last terms then we will get
$3.5=15....\left( \text{iv} \right)$
According to FOIL rule the product of the two functions we will obtained by adding the results of equations $\left( \text{i} \right)$, $\left( \text{ii} \right)$, $\left( \text{iii} \right)$, $\left( \text{iv} \right)$ , then we will get
$\left( x+3 \right)\left( x+5 \right)={{x}^{2}}+5x+3x+15$
We know that $ax+bx=\left( a+b \right)x$, then we will get
$\begin{align}
  & \Rightarrow \left( x+3 \right)\left( x+5 \right)={{x}^{2}}+\left( 5+3 \right)x+15 \\
 & \therefore \left( x+3 \right)\left( x+5 \right)={{x}^{2}}+8x+15 \\
\end{align}$

Note:
In the problem they have asked to use the FOIL rule. If they don’t mention the method, they can multiply the terms directly. Then we will get
$\left( x+3 \right)\left( x+5 \right)=x\left( x+5 \right)+3\left( x+5 \right)$
Applying multiplication distribution law in the above equation, then we will get
$\begin{align}
  & \Rightarrow \left( x+3 \right)\left( x+5 \right)=x.x+x.5+3.x+3.5 \\
 & \Rightarrow \left( x+3 \right)\left( x+5 \right)={{x}^{2}}+5x+3x+15 \\
 & \therefore \left( x+3 \right)\left( x+5 \right)={{x}^{2}}+8x+15 \\
\end{align}$
From both the methods we got the same result.