Answer
See explanation
Work Step by Step
Denote the first number as $M$, so that
\begin{equation}
M= 5+i\sqrt{15}
\end{equation}
and the second number as $N$, so that
\begin{equation}
N= 5-i\sqrt{15}
\end{equation}
We want to proof that the sum of the numbers, $M$ and $N$ is equal to 10 and their product is equal to 40. First add the numbers to get the required sum. Then, multiply them to get the product.
\begin{equation}
\begin{aligned}
M+N&=5+i\sqrt{15}+5-i\sqrt{15} \\
&=(5+5)+i(\sqrt{15}-\sqrt{15})\\
&= 10+i(0)\\
&= 10\\
M\cdot N&=(5+i\sqrt{15}) \cdot (5-i\sqrt{15})\\
&= (5+i\sqrt{15}) 5-i\sqrt{15}(5+i\sqrt{15})\\
&= 5^2+i\sqrt{15} -i\sqrt{15}-(i\sqrt{15})^2\\
&= 25-15i^2\\
&= 25-15(-1)\\
&= 25+15\\
&= 40
\end{aligned}
\end{equation}
Thus, we have proved the condition stated in the problem.
