Answer is: option3

\( y = \frac{5}{2}x - \frac{1}{2} \)

Solution:

We need to find the equation of the tangent line at the point \( (1,2) \).

Differentiate \( y = x\sqrt{3 + x^2} \)

Now applying the product rule:

\[ \frac{dy}{dx} = 1 \cdot \sqrt{3 + x^2} + x \cdot \frac{x}{\sqrt{3 + x^2}} \] \[ \frac{dy}{dx} = \sqrt{3 + x^2} + \frac{x^2}{\sqrt{3 + x^2}} \] \[ \frac{dy}{dx} = \frac{(3 + x^2) + x^2}{\sqrt{3 + x^2}} \] \[ \frac{dy}{dx} = \frac{3 + 2x^2}{\sqrt{3 + x^2}} \]

Substituting \( x = 1 \):

\[ \frac{dy}{dx} = \frac{3 + 2(1)^2}{\sqrt{3 + 1^2}} \] \[ = \frac{3 + 2}{\sqrt{4}} \] \[ = \frac{5}{2} \]

So, the slope of the tangent line is \( \frac{5}{2} \).

Using the point-slope formula: \[ y - y_1 = m(x - x_1) \]

Substituting \( (x_1, y_1) = (1,2) \) and \( m = \frac{5}{2} \):

\[ y - 2 = \frac{5}{2} (x - 1) \] \[ y = \frac{5}{2}x - \frac{5}{2} + 2 \] \[ y = \frac{5}{2}x - \frac{1}{2} \]

The correct answer is:

\[ \boxed{y = \frac{5}{2}x - \frac{1}{2}} \]