Derivative of f(x)=x^2(3x+x^3)

1. Apply the product rule:

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

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

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

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

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

      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: $3$

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

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

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

2. Now simplify:

   $$x^{2} \cdot \left(5 x^{2} + 9\right)$$

The answer is:

$$x^{2} \cdot \left(5 x^{2} + 9\right)$$