The given function is:

$f(x) = \left(-4x^2 - 5x + 7 \right) \cdot \frac{-2x - 1}{-4x^2 - 5x + 7}$

We need to compute its derivative, $f'(x)$.

Step-by-Step Solution

1. Simplify the function: Notice that the polynomial $(-4x^2 - 5x + 7)$ appears in both the numerator and the denominator of the product. So, we can simplify the entire function:

$f(x) = (-2x - 1)$

2. Differentiate: Now, take the derivative of $f(x) = -2x - 1$ using basic differentiation rules:

$f'(x) = \frac{d}{dx}(-2x - 1) = -2$

Thus, the derivative of the given function is:

$f'(x) = -2$

Would you like more details on any step? Here are five related questions you could explore:

1. What happens if the function had more complicated terms that couldn’t be simplified?
2. How do you apply the product and quotient rules for differentiation?
3. Why does the derivative of a constant term like $-1$ become zero?
4. How does the chain rule apply when differentiating more complex functions?
5. Can we use implicit differentiation for functions like this?

Tip: Always check for any simplifications before applying differentiation rules to avoid unnecessary complexity!