The polynomial f(x) is given by f(x) = x^3 + 6x^2 + x - 4. Hence, find the exact roots of the equation f(x) = 0.

To find the roots of the equation f(x) = 0, we need to solve the equation x^3 + 6x^2 + x - 4 = 0.

One way to find the roots is by factoring the polynomial. However, in this case, the polynomial is not easily factorable. Therefore, we can use numerical methods such as the Newton-Raphson method or synthetic division to approximate the roots.

Using the Newton-Raphson method, we start with an initial guess for the root and iterate using the formula:

x_(n+1) = x_n - f(x_n) / f'(x_n)

where f'(x) is the derivative of the function f(x).

By iterating this formula, we can find the roots of the equation f(x) = 0.

Step 1: Factor the polynomial using Rational Root Theorem

- List the factors of the constant term (-4): ±1, ±2, ±4
- List the factors of the leading coefficient (1): ±1
- Try all possible combinations of the factors to find rational roots.

We find that x = 2 is a rational root.

Step 2: Perform Synthetic Division

```
2 | 1 6 1 -4
\ 2 8 18
---
1 8 9 14
```

The last number in the bottom row (14) is the remainder, which is not zero. Therefore, x = 2 is not a root of f(x) = 0.

Step 3: Use the Rational Root Theorem Again

Since 2 is not a root, the numerator of the rational root must be a factor of 14. The factors of 14 are: ±1, ±2, ±7, ±14

Repeat the synthetic division process for each factor of 14. We find that x = -7 is a rational root.

Step 4: Factor the Reduced Polynomial

Using synthetic division, we divide f(x) by (x - (-7)):

```
-7 | 1 6 1 -4
\ -7 35 -245
---
1 -1 36 -249
```

The reduced polynomial is x^2 - x + 36. This is an irreducible quadratic, which means it cannot be further factored over real numbers.

Step 5: Find the Roots of the Irreducible Quadratic

Using the quadratic formula, we can find the roots of x^2 - x + 36 = 0:

```
x = (1 ± √(1 - 4 - 1 - 36)) / 2 - 1
x = (1 ± √(-143)) / 2
```

Since there are no real solutions to the quadratic, the roots of f(x) = 0 are:

x = -7 (rational root)

x = (1 ± √(-143)) / 2 (complex conjugate roots)