Derivative of y=(1+x²)(3-2x)

1. Apply the product rule:

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

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

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

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

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

      The result is: $2 x$

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

   1. Differentiate $3 - 2 x$ term by term:

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

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

         1. 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: $2$

         So, the result is: $-2$

      The result is: $-2$

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

2. Now simplify:

   $$- 6 x^{2} + 6 x - 2$$

The answer is:

$$- 6 x^{2} + 6 x - 2$$