Derivative of (x²+3x-2)²

1. Let $u = \left(x^{2} + 3 x\right) - 2$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x^{2} + 3 x\right) - 2\right)$:

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

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

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

         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: $3$

         The result is: $2 x + 3$

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

      The result is: $2 x + 3$

   The result of the chain rule is:

   $$\left(2 x + 3\right) \left(2 \left(x^{2} + 3 x\right) - 4\right)$$

4. Now simplify:

   $$2 \left(2 x + 3\right) \left(x^{2} + 3 x - 2\right)$$

The answer is:

$$2 \left(2 x + 3\right) \left(x^{2} + 3 x - 2\right)$$