Derivative of f(x)=(5x²+7x)(4x³-3)

1. Apply the product rule:

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

   $f{\left(x \right)} = 5 x^{2} + 7 x$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $5 x^{2} + 7 x$ 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: $x^{2}$ goes to $2 x$

         So, the result is: $10 x$

      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: $7$

      The result is: $10 x + 7$

   $g{\left(x \right)} = 4 x^{3} - 3$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $4 x^{3} - 3$ 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: $x^{3}$ goes to $3 x^{2}$

         So, the result is: $12 x^{2}$

      2. The derivative of the constant $\left(-1\right) 3$ is zero.

      The result is: $12 x^{2}$

   The result is: $12 x^{2} \cdot \left(5 x^{2} + 7 x\right) + \left(10 x + 7\right) \left(4 x^{3} - 3\right)$

2. Now simplify:

   $$100 x^{4} + 112 x^{3} - 30 x - 21$$

The answer is:

$$100 x^{4} + 112 x^{3} - 30 x - 21$$