Answer
$3(s+5i)(s-5i)$
Work Step by Step
The given expression, $ 3s^2+75 ,$ is equivalent to \begin{align*} & 3(s^2+25) \\&= 3[s^2-(-25)] \\&= 3[s^2-(-1\cdot25)] .\end{align*} Since $i^2=-1,$ the expression above is equivalent to \begin{align*} & 3[s^2-(i^2\cdot25)] \\&= 3[s^2-(25i^2)] .\end{align*} Using $a^2-b^2=(a+b)(a-b)$ or the factoring of the difference of $2$ squares, the expression above is equivalent to \begin{align*} & 3[(s)^2-(5i)^2] \\&= 3[(s+5i)(s-5i)] \\&= 3(s+5i)(s-5i) .\end{align*} Hence, the factored form of the given expression is $
3(s+5i)(s-5i)
.$
Checking: Multiplying the factors above results to
\begin{align*}
&
3[(s+5i)(s-5i)]
\\&=
3[(s)^2-(5i)^2]
&\left(\text{use }(a+b)(a-b)=a^2-b^2\right)
\\&=
3(s^2-25i^2)
\\&=
3(s^2)-3(25i^2)
&\left(\text{use Distributive Property}\right)
\\&=
3s^2-75i^2
\\&=
3s^2-75(-1)
&\left(\text{use }i^2=-1\right)
\\&=
3s^2+75
\text{ (same as original expression)}
.\end{align*}