Derivative of ((x³-2x²+3)(x⁴+x+1))

1. Apply the product rule:

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

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

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

      1. Differentiate $x^{3} - 2 x^{2}$ 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: $- 4 x$

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

      2. The derivative of the constant $3$ is zero.

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

   $g{\left(x \right)} = \left(x^{4} + x\right) + 1$; to find $\frac{d}{d x} g{\left(x \right)}$:

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

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

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

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

         The result is: $4 x^{3} + 1$

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

      The result is: $4 x^{3} + 1$

   The result is: $\left(3 x^{2} - 4 x\right) \left(\left(x^{4} + x\right) + 1\right) + \left(4 x^{3} + 1\right) \left(\left(x^{3} - 2 x^{2}\right) + 3\right)$

2. Now simplify:

   $$7 x^{6} - 12 x^{5} + 16 x^{3} - 3 x^{2} - 4 x + 3$$

The answer is:

$$7 x^{6} - 12 x^{5} + 16 x^{3} - 3 x^{2} - 4 x + 3$$