8. Given $f(x)=1-2 x^{2}$, find $\frac{f(x+h)-f(x)}{h}$
A. -9
C. $-4 x-2 h$
B. $2 x+h$
D. $\frac{4 x^{2}+2 x h+h^{2}}{h}$

Step 1 ：The problem is asking for the difference quotient of the function \(f(x)=1-2 x^{2}\). The difference quotient is a measure of the average rate of change of the function over the interval \([x, x+h]\). It is a fundamental concept in calculus and is used to define the derivative of a function.

Step 2 ：To find the difference quotient, we need to substitute \(x+h\) into the function, subtract the function at \(x\), and then divide by \(h\).

Step 3 ：Substitute \(x+h\) into the function: \(f(x+h) = 1 - 2*(x+h)^{2}\)

Step 4 ：Subtract the function at \(x\): \(f(x+h) - f(x) = 1 - 2*(x+h)^{2} - (1 - 2*x^{2})\)

Step 5 ：Divide by \(h\) to get the difference quotient: \(\frac{f(x+h)-f(x)}{h} = -2h - 4x\)

Step 6 ：The simplified form of the expression \(\frac{f(x+h)-f(x)}{h}\) for the function \(f(x)=1-2 x^{2}\) is \(-2h - 4x\).

Step 7 ：Final Answer: The difference quotient of the function \(f(x)=1-2 x^{2}\) is \(\boxed{-4x - 2h}\)