Derivative of f(x)=(x²+3)(x-5)

1. Apply the product rule:

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

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

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

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

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

      The result is: $2 x$

   $g{\left(x \right)} = x - 5$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $x - 5$ term by term:

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

      2. The derivative of the constant $-5$ is zero.

      The result is: $1$

   The result is: $x^{2} + 2 x \left(x - 5\right) + 3$

2. Now simplify:

   $$3 x^{2} - 10 x + 3$$

The answer is:

$$3 x^{2} - 10 x + 3$$