1. Problem: Given the equation $$2x + \frac{2}{x} = 3$$, find the value of $$x^2 + \text{?}$$ where the options are \(\frac{1}{6}, \frac{1}{4}, \frac{1}{3}, \frac{1}{5}\). 2. Formula and rules: We want to find $$x^2 + \frac{1}{x^2}$$. We start from the given equation and use algebraic identities. 3. Start with the given equation: $$2x + \frac{2}{x} = 3$$ Divide both sides by 2: $$x + \frac{1}{x} = \frac{3}{2}$$ 4. Square both sides to use the identity: $$\left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2}$$ 5. Substitute the value: $$\left(\frac{3}{2}\right)^2 = x^2 + 2 + \frac{1}{x^2}$$ $$\frac{9}{4} = x^2 + 2 + \frac{1}{x^2}$$ 6. Rearrange to find $$x^2 + \frac{1}{x^2}$$: $$x^2 + \frac{1}{x^2} = \frac{9}{4} - 2 = \frac{9}{4} - \frac{8}{4} = \frac{1}{4}$$ 7. The question asks for $$x^2 + \text{?}$$ which corresponds to $$x^2 + \frac{1}{x^2}$$, so the answer is $$\frac{1}{4}$$. 8. However, the provided correct answer is option গ) $$\frac{1}{3}$$, so let's verify if the question might be asking for $$x^2 + k$$ where $$k = \frac{1}{x^2}$$ and the value is $$\frac{1}{3}$$. 9. Since our calculation shows $$\frac{1}{4}$$, the closest correct answer from the options is খ) $$\frac{1}{4}$$. Final answer: $$\boxed{\frac{1}{4}}$$