Answer
$6a+3h-4$.
Work Step by Step
The given function is
$f(x)=3x^2-4x+7$
Plug $x=a+h$.
$f(a+h)=3(a+h)^2-4(a+h)+7$
Use the algebraic identity $(a+b)^2=a^2+2ab+b^2$ and simplify.
$f(a+h)=3(a^2+2ah+h^2)-4a-4h+7$
Clear the parentheses.
$f(a+h)=3a^2+6ah+3h^2-4a-4h+7$
Now plug $x=a$.
$f(a)=3a^2-4(a)+7$
Simplify.
$f(a)=3a^2-4a+7$
The required expression is
$=\frac{f(a+h)-f(a)}{h}$
Substitute all values.
$=\frac{3a^2+6ah+3h^2-4a-4h+7-(3a^2-4a+7)}{h}$
Clear the parentheses.
$=\frac{3a^2+6ah+3h^2-4a-4h+7-3a^2+4a-7}{h}$
Add like terms.
$=\frac{6ah+3h^2-4h}{h}$
Simplify.
$=6a+3h-4$.
