Answer
$a)$ $(f+g)(x)=x^{2}+5x+1$
$b)$ $(f-g)(x)=x^{2}-5x+1$
$c)$ $(f\cdot g)(x)=5x^{3}+5x$
$d)$ $\Big(\dfrac{f}{g}\Big)(x)=\dfrac{x^{2}+1}{5x}$, where $x\ne0$
Work Step by Step
$f(x)=x^{2}+1;$ $g(x)=5x$
For each case, replace $f(x)$ by $x^{2}+1$ and $g(x)$ by $5x$ and simplify if possible
$a)$ $(f+g)(x)$
$f(x)+g(x)=(x^{2}+1)+(5x)=x^{2}+1+5x=x^{2}+5x+1$
$b)$ $(f-g)(x)$
$f(x)-g(x)=(x^{2}+1)-(5x)=x^{2}+1-5x=x^{2}-5x+1$
$c)$ $(f\cdot g)(x)$
$f(x)\cdot g(x)=(x^{2}+1)(5x)=5x^{3}+5x$
$d)$ $\Big(\dfrac{f}{g}\Big)(x)$
$\dfrac{f(x)}{g(x)}=\dfrac{x^{2}+1}{5x}$, where $x\ne0$
