We are asked to evaluate the definite integral $\int_{-1}^{3} (5x^2 - x - 4) dx$.

1. Problem Description

We are asked to evaluate the definite integral $\int_{-1}^{3} (5x^2 - x - 4) dx$.

2. Solution Steps

First, we find the antiderivative of the integrand $5x^2 - x - 4$.

Using the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, we get:

$$\int (5x^2 - x - 4) dx = 5 \int x^2 dx - \int x dx - 4 \int 1 dx$$

$$= 5 \frac{x^3}{3} - \frac{x^2}{2} - 4x + C = \frac{5}{3}x^3 - \frac{1}{2}x^2 - 4x + C$$

Now, we evaluate the definite integral:

$$\int_{-1}^{3} (5x^2 - x - 4) dx = \left[\frac{5}{3}x^3 - \frac{1}{2}x^2 - 4x\right]_{-1}^{3}$$

$$= \left(\frac{5}{3}(3)^3 - \frac{1}{2}(3)^2 - 4(3)\right) - \left(\frac{5}{3}(-1)^3 - \frac{1}{2}(-1)^2 - 4(-1)\right)$$

$$= \left(\frac{5}{3}(27) - \frac{1}{2}(9) - 12\right) - \left(\frac{5}{3}(-1) - \frac{1}{2}(1) + 4\right)$$

$$= \left(45 - \frac{9}{2} - 12\right) - \left(-\frac{5}{3} - \frac{1}{2} + 4\right)$$

$$= 33 - \frac{9}{2} + \frac{5}{3} + \frac{1}{2} - 4$$

$$= 29 - \frac{8}{2} + \frac{5}{3} = 29 - 4 + \frac{5}{3} = 25 + \frac{5}{3}$$

$$= \frac{75}{3} + \frac{5}{3} = \frac{80}{3}$$

3. Final Answer

$\frac{80}{3}$