Derivative of x²(x+1)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 $2 x$

   $g{\left(x \right)} = \left(x + 1\right)^{3}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = x + 1$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$:

      1. Differentiate $x + 1$ term by term:

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

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

         The result is: $1$

      The result of the chain rule is:

      $$3 \left(x + 1\right)^{2}$$

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

2. Now simplify:

   $$x \left(x + 1\right)^{2} \cdot \left(5 x + 2\right)$$

The answer is:

$$x \left(x + 1\right)^{2} \cdot \left(5 x + 2\right)$$