The given mathematical expression in the image is:

$\left( \frac{3 + \sqrt{13}}{2} \right)^2 - 3 \left( \frac{3 + \sqrt{13}}{2} \right) - 1 = 0$

Let's denote $x = \frac{3 + \sqrt{13}}{2}$. Substituting this into the expression, we have:

$x^2 - 3x - 1 = 0$

This is a standard quadratic equation in terms of $x$. To solve it, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 1$, $b = -3$, and $c = -1$. Plugging these values in:

$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}$

$x = \frac{3 \pm \sqrt{9 + 4}}{2}$

$x = \frac{3 \pm \sqrt{13}}{2}$

So, the solutions are:

$x_1 = \frac{3 + \sqrt{13}}{2}, \quad x_2 = \frac{3 - \sqrt{13}}{2}$

Since we defined $x = \frac{3 + \sqrt{13}}{2}$, it means $x_1 = x$ matches our original substitution, which confirms that $x = \frac{3 + \sqrt{13}}{2}$ is indeed a solution of the equation.

Thus, the equation is satisfied with the value $x = \frac{3 + \sqrt{13}}{2}$.

Would you like more details on the solution or have any specific questions? Here are some related questions you might find interesting:

1. How do you derive the quadratic formula?
2. What is the significance of the discriminant in a quadratic equation?
3. Can we factorize this quadratic equation directly without using the quadratic formula?
4. How do we interpret the solutions of a quadratic equation graphically?
5. What happens if the discriminant of a quadratic equation is zero or negative?

Tip: When solving quadratic equations, always check if you can simplify or factorize the expression before applying the quadratic formula for efficiency.