The function you've provided is:

$f(x) = -4x^3 + 5x^2 - 6x + 1$

To find $4f(x)$, we simply multiply the entire expression for $f(x)$ by 4:

$4f(x) = 4 \times \left( -4x^3 + 5x^2 - 6x + 1 \right)$

Now distribute the 4 to each term:

$4f(x) = -16x^3 + 20x^2 - 24x + 4$

So, the expression for $4f(x)$ is:

$4f(x) = -16x^3 + 20x^2 - 24x + 4$

Would you like more details or have any questions about this? Here are 5 related questions you might find interesting:

1. How do you factor a cubic polynomial like $-4x^3 + 5x^2 - 6x + 1$?
2. What is the derivative of $f(x)$ with respect to $x$?
3. How do you evaluate $f(x)$ at a specific value of $x$, like $x = 2$?
4. What happens when you multiply a polynomial by a constant like 4?
5. How would you find the roots or solutions of the equation $f(x) = 0$?

Tip: When working with polynomials, always be mindful of how terms behave when multiplied by a constant—each term in the polynomial gets scaled.