Derivative of y=(x²+3x)(x+1)

1. Apply the product rule:

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

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

   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: $2 x + 3$

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

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

2. Now simplify:

   $$3 x^{2} + 8 x + 3$$

The answer is:

$$3 x^{2} + 8 x + 3$$