Answer
(a.) $f(a+2)=a^2+a+5$.
(b.) $f(a+h)-f(a)=2ah+h^2-3h$.
Work Step by Step
The given function is
$f(x)=x^2-3x+7$
(a.) Plug $x=a+2$ into the given function.
$f(a+2)=(a+2)^2-3(a+2)+7$
Clear the parentheses.
$f(a+2)=a^2+4a+4-3a-6+7$
Simplify.
$f(a+2)=a^2+a+5$
(b.) Plug $x=a+h$ into the given function.
$f(a+h)=(a+h)^2-3(a+h)+7$
Clear the parentheses.
$f(a+h)=a^2+2ah+h^2-3a-3h+7$
Simplify.
$f(a+h)=a^2+2ah+h^2-3a-3h+7$ ... (1)
Now plug $x=a$ into the given function.
$f(a)=(a)^2-3(a)+7$
Clear the parentheses.
$f(a)=a^2-3a+7$ ...(2)
Subtract function (2) from function (1).
$f(a+h)-f(a)=(a^2+2ah+h^2-3a-3h+7)-(a^2-3a+7)$
Clear the parentheses.
$f(a+h)-f(a)=a^2+2ah+h^2-3a-3h+7-a^2+3a-7$
Simplify.
$f(a+h)-f(a)=2ah+h^2-3h$.
