Derivative of (x³-2)(x²+1)

1. Apply the product rule:

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

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

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

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

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

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

   $g{\left(x \right)} = x^{2} + 1$; to find $\frac{d}{d x} g{\left(x \right)}$:

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

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

      2. The derivative of the constant $1$ is zero.

      The result is: $2 x$

   The result is: $3 x^{2} \left(x^{2} + 1\right) + 2 x \left(x^{3} - 2\right)$

2. Now simplify:

   $$x \left(5 x^{3} + 3 x - 4\right)$$

The answer is:

$$x \left(5 x^{3} + 3 x - 4\right)$$