For the function $$ f(x)=4 x^{2}-3 x+3 $$, evaluate and simplify the difference quotient. $$ \frac{f(x+h)-f(x)}{h}=\square $$

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To evaluate and simplify the difference quotient for the function $$ f(x) = 4x^2 - 3x + 3 $$, follow these steps:

1. **Compute $$ f(x+h) $$:**
   $$
   f(x+h) = 4(x+h)^2 - 3(x+h) + 3
   $$
   Expand the terms:
   $$
   = 4(x^2 + 2xh + h^2) - 3x - 3h + 3
   $$
   $$
   = 4x^2 + 8xh + 4h^2 - 3x - 3h + 3
   $$

2. **Find $$ f(x+h) - f(x) $$:**
   $$
   f(x+h) - f(x) = (4x^2 + 8xh + 4h^2 - 3x - 3h + 3) - (4x^2 - 3x + 3)
   $$
   Simplify by subtracting like terms:
   $$
   = 8xh + 4h^2 - 3h
   $$

3. **Divide by $$ h $$ to get the difference quotient:**
   $$
   \frac{f(x+h) - f(x)}{h} = \frac{8xh + 4h^2 - 3h}{h}
   $$
   Simplify by cancelling $$ h $$:
   $$
   = 8x + 4h - 3
   $$

**Final Answer:**
$$
\frac{f(x+h) - f(x)}{h} = 8x + 4h - 3
$$
$$ \frac{f(x+h) - f(x)}{h} = 8x + 4h - 3 $$

For the function , evaluate and simplify the difference quotient.

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