Answer
(a.) $f(a+2)=4a^2+21a+25$.
(b.) $f(a+h)-f(a)=8ah+4h^2+5h$.
Work Step by Step
The given function is
$f(x)=4x^2+5x-1$
(a.) Plug $x=a+2$ into the given function.
$f(a+2)=4(a+2)^2+5(a+2)-1$
Clear the parentheses.
$f(a+2)=4a^2+16a+16+5a+10-1$
Simplify.
$f(a+2)=4a^2+21a+25$.
(b.) Plug $x=a+h$ into the given function.
$f(a+h)=4(a+h)^2+5(a+h)-1$
Clear the parentheses.
$f(a+h)=4a^2+8ah+4h^2+5a+5h-1$... (1)
Now plug $x=a$ into the given function.
$f(a)=4(a)^2+5(a)-1$
Clear the parentheses.
$f(a)=4a^2+5a-1$ ...(2)
Subtract function (2) from function (1).
$f(a+h)-f(a)=(4a^2+8ah+4h^2+5a+5h-1)-(4a^2+5a-1)$
Clear the parentheses.
$f(a+h)-f(a)=4a^2+8ah+4h^2+5a+5h-1-4a^2-5a+1$
Simplify.
$f(a+h)-f(a)=8ah+4h^2+5h$.
