Your answer is incorrect.
Find the difference quotient $\frac{f(x+h)-f(x)}{h}$, where $h \neq 0$, for the function below.
\[
f(x)=-4 x^{2}-2 x+1
\]
Simplify your answer as much as possible.
\[
\frac{f(x+h)-f(x)}{h}=
\]
Check
Final Answer: The simplified difference quotient for the function \(f(x) = -4x^2 - 2x + 1\) is \(\boxed{-4h - 8x - 2}\).
Step 1 :The difference quotient is a measure of the average rate of change of the function over the interval h. To find the difference quotient for the function \(f(x) = -4x^2 - 2x + 1\), we need to substitute \(x+h\) into the function for \(x\), subtract the original function, and then divide by \(h\).
Step 2 :Substitute \(x+h\) into the function for \(x\): \(f(x+h) = -4(x+h)^2 - 2(x+h) + 1\).
Step 3 :Subtract the original function: \(f(x+h) - f(x) = -4(x+h)^2 - 2(x+h) + 1 - (-4x^2 - 2x + 1)\).
Step 4 :Divide by \(h\) to get the difference quotient: \(\frac{f(x+h)-f(x)}{h} = \frac{-4(x+h)^2 - 2(x+h) + 1 - (-4x^2 - 2x + 1)}{h}\).
Step 5 :Simplify the difference quotient to get the final answer: \(\frac{f(x+h)-f(x)}{h} = -4h - 8x - 2\).
Step 6 :Final Answer: The simplified difference quotient for the function \(f(x) = -4x^2 - 2x + 1\) is \(\boxed{-4h - 8x - 2}\).