Given the function $f(x)=-1-4 x$, express the value of $\frac{f(x+h)-f(x)}{h}$ in simplest form.

Step 1 ：The problem is asking for the difference quotient of the function \(f(x) = -1 - 4x\). The difference quotient is a formula used in calculus to find the derivative of a function. It is defined as \(\frac{f(x+h)-f(x)}{h}\).

Step 2 ：To find the difference quotient, we need to substitute \(x+h\) into the function \(f(x)\) and then subtract \(f(x)\) from it. After that, we divide the result by \(h\).

Step 3 ：Substituting \(x+h\) into the function \(f(x)\), we get \(f(x+h) = -4*(x+h) - 1 = -4x - 4h - 1\).

Step 4 ：Subtracting \(f(x)\) from \(f(x+h)\), we get \(f(x+h) - f(x) = -4x - 4h - 1 - (-4x - 1) = -4h\).

Step 5 ：Dividing the result by \(h\), we get \(\frac{f(x+h)-f(x)}{h} = \frac{-4h}{h} = -4\).

Step 6 ：So, the value of \(\frac{f(x+h)-f(x)}{h}\) for the function \(f(x) = -1 - 4x\) is -4.

Step 7 ：Final Answer: The value of \(\frac{f(x+h)-f(x)}{h}\) in simplest form is \(\boxed{-4}\).