If $$\alpha$$ and $$\beta$$ are the roots of the equation $$2x(2x+1) = 1$$, then $$\beta$$ is equal to:

1. Simplify the Quadratic Equation

The given equation is: 

$$2x(2x + 1) = 1$$ Expand the left side:

$$4x^2 + 2x = 1$$

$$4x^2 + 2x - 1 = 0$$

2. Relationship between Roots ($$\alpha$$ and $$\beta$$)

For a quadratic equation of the form $$ax^2 + bx + c = 0$$, the sum and product of roots are given by:

• Sum of roots: $$\displaystyle \alpha + \beta = -\frac{b}{a} = -\frac{2}{4} = -\frac{1}{2}$$
• Product of roots: $$\displaystyle \alpha\beta = \frac{c}{a} = -\frac{1}{4}$$

From the sum of roots, we can express $$\beta$$ in terms of $$\alpha$$:

$$\beta = -\frac{1}{2} - \alpha$$

3. Expressing $$\beta$$ in the Form of the Options

We need to check which option matches $$\beta = -\frac{1}{2} - \alpha$$. Let's test the expression in Option B:

$$-2\alpha(\alpha + 1) = -2\alpha^2 - 2\alpha$$

Since $\alpha$ is a root of the equation $$4x^2 + 2x - 1 = 0$$, it must satisfy the equation:

$$4\alpha^2 + 2\alpha - 1 = 0$$

Rearranging to find $$2\alpha^2$$:

$$4\alpha^2 = 1 - 2\alpha$$

$$2\alpha^2 = \frac{1 - 2\alpha}{2} = \frac{1}{2} - \alpha$$

Now, substitute this value of $$2\alpha^2$$ into our expression for Option B:

$$-2\alpha^2 - 2\alpha = -\left( \frac{1}{2} - \alpha \right) - 2\alpha$$

$$= -\frac{1}{2} + \alpha - 2\alpha$$

$$= -\frac{1}{2} - \alpha$$

Final Result

Since the expression in Option B simplifies exactly to $$-\frac{1}{2} - \alpha$$, which we found to be the value of $$\beta$$:

$$\beta = -2\alpha(\alpha + 1)$$

The correct option is B.