Derivative of x^4+4*x^3-8*x^2-5

1. Differentiate $\left(- 8 x^{2} + \left(x^{4} + 4 x^{3}\right)\right) - 5$ term by term:

   1. Differentiate $- 8 x^{2} + \left(x^{4} + 4 x^{3}\right)$ term by term:

      1. Differentiate $x^{4} + 4 x^{3}$ term by term:

         1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$

         2. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

            So, the result is: $12 x^{2}$

         The result is: $4 x^{3} + 12 x^{2}$

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x^{2}$ goes to $2 x$

         So, the result is: $- 16 x$

      The result is: $4 x^{3} + 12 x^{2} - 16 x$

   2. The derivative of the constant $-5$ is zero.

   The result is: $4 x^{3} + 12 x^{2} - 16 x$

2. Now simplify:

   $$4 x \left(x^{2} + 3 x - 4\right)$$

The answer is:

$$4 x \left(x^{2} + 3 x - 4\right)$$