Derivative of y=(x^3+3x^2)(x^2-8)

1. Apply the product rule:

   $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

   $f{\left(x \right)} = x^{3} + 3 x^{2}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $x^{3} + 3 x^{2}$ term by term:

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

      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: $6 x$

      The result is: $3 x^{2} + 6 x$

   $g{\left(x \right)} = x^{2} - 8$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $x^{2} - 8$ term by term:

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

      2. The derivative of the constant $\left(-1\right) 8$ is zero.

      The result is: $2 x$

   The result is: $2 x \left(x^{3} + 3 x^{2}\right) + \left(x^{2} - 8\right) \left(3 x^{2} + 6 x\right)$

2. Now simplify:

   $$x \left(5 x^{3} + 12 x^{2} - 24 x - 48\right)$$

The answer is:

$$x \left(5 x^{3} + 12 x^{2} - 24 x - 48\right)$$