Find the difference quotient $\frac{f(x+h)-f(x)}{h}$, where $h \neq 0$, for the function below. \[ f(x)=-3 x^{2}-1 \]

Step 1 ：Given the function \(f(x)=-3 x^{2}-1\), we are asked to find the difference quotient \(\frac{f(x+h)-f(x)}{h}\), where \(h \neq 0\).

Step 2 ：The difference quotient is a measure of the average rate of change of the function over the interval \(h\). It is a fundamental concept in calculus and is the basis for the definition of the derivative.

Step 3 ：To find the difference quotient for the function, we need to substitute \(x+h\) into the function, subtract the function at \(x\), and then divide by \(h\).

Step 4 ：Substituting \(x+h\) into the function, we get \(-3(x+h)^{2}-1\).

Step 5 ：Subtracting the function at \(x\), we get \(-3(x+h)^{2}-1 - (-3x^{2}-1)\).

Step 6 ：Simplifying the above expression, we get \(-3h - 6x\).

Step 7 ：Finally, dividing by \(h\), we get the difference quotient as \(-3 - 6x\).

Step 8 ：Final Answer: The difference quotient for the function \(f(x)=-3 x^{2}-1\) is \(\boxed{-3 - 6x}\).