Solving the given equation using Rational Root Theorem:
The Rational Root Theorem states that if a polynomial equation has a rational root (i.e. a root that can be expressed as a fraction), then that root will be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

In this case, the constant term is -2 and the leading coefficient is 4. Therefore, any rational root of the equation must be of the form p/q, where p is a factor of -2 and q is a factor of 4.

The factors of -2 are -1, 1, -2, and 2. The factors of 4 are -1, 1, -2, 2, -4, and 4. Therefore, the possible rational roots of the equation are:

±1/1, ±2/1, ±1/2, ±2/2, ±4/1, and ±4/2

Simplifying these fractions, we get:

±1, ±2, ±1/2, ±1, ±4, and ±2

Testing the possible rational roots:
We can now test each of these possible rational roots by substituting them into the given equation and checking if they satisfy the equation.

For example, if we substitute x = 1 into the equation, we get:

4(1)^3 + 8(1)^2 - 1 - 2 = 0

which simplifies to:

4 + 8 - 1 - 2 = 0

Therefore, x = 1 is a root of the equation.

Similarly, we can test the other possible rational roots and find that x = -1/2 and x = -2 are also roots of the equation.

Using synthetic division to factor the equation:
Now that we have found three roots of the equation, we can use synthetic division to factor the equation.

Using synthetic division with x = 1 as the divisor, we get:

1 | 4 8 -1 -2
| 4 12 11
|---------
| 4 12 10 9

Therefore, we can write the equation as:

(x - 1)(4x^2 + 12x + 10x + 9) = 0

Simplifying the second factor, we get:

(x - 1)(4x^2 + 22x + 9) = 0

Using the quadratic formula to solve for the roots of the second factor, we get:

x = (-22 ± sqrt(22^2 - 4(4)(9))) / (2(4))

which simplifies to:

x = (-22 ± sqrt(436)) / 8

x ≈ -2.24 and x ≈ -0.51

Therefore, the four roots of the equation are:

x = 1, x = -1/2, x = -2.24, and x = -0.51

Using the roots to find the value of (2x + 3):
We are asked to find the value of (2x + 3) when x = 1, x = -1/2, and x = -2.24.

When x