To solve the problem, we are given the function:

$f(x) = 2 + 4x - 5x^2$

and asked to find $f'(-4)$, which is the value of the derivative of the function at $x = -4$.

Steps:

1. Differentiate the function $f(x)$ with respect to $x$:

   $f'(x) = \frac{d}{dx} \left( 2 + 4x - 5x^2 \right)$

   • The derivative of a constant (2) is 0.
   • The derivative of $4x$ is 4.
   • The derivative of $-5x^2$ is $-10x$ (using the power rule).

   Thus, the derivative is:

   $f'(x) = 4 - 10x$

2. Substitute $x = -4$ into the derivative:

   $f'(-4) = 4 - 10(-4)$ $f'(-4) = 4 + 40 = 44$

Final Answer:

$f'(-4) = 44$

Would you like any further details on this problem?

Here are 5 related questions you might want to consider:

1. How do you apply the power rule for differentiation?
2. What are the steps for differentiating polynomial functions?
3. How do you find the second derivative of a function?
4. How do derivatives relate to the slope of the tangent line at a point?
5. Can you interpret the physical meaning of a derivative in real-world problems?

Tip: Always check your derivative calculations by simplifying each term carefully before evaluating at a specific point.