Given the function $f(x)=-1-4x$, 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 $f(x+h)=-4\ast (x+h)-1=-4x-4h-1$.

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 $\boxed{{\displaystyle -4}}$.