Derivative of y=(x²+2)(x³-x)

1. Apply the product rule:

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

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

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

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

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

      The result is: $2 x$

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

   1. Differentiate $x^{3} - x$ 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: $x$ goes to $1$

         So, the result is: $-1$

      The result is: $3 x^{2} - 1$

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

2. Now simplify:

   $$5 x^{4} + 3 x^{2} - 2$$

The answer is:

$$5 x^{4} + 3 x^{2} - 2$$