(x-3)(x+1)=9/4 equation

Move right part of the equation to
left part with negative sign.

The equation is transformed from
$$\left(x - 3\right) \left(x + 1\right) = \frac{9}{4}$$
to
$$\left(x - 3\right) \left(x + 1\right) - \frac{9}{4} = 0$$
Expand the expression in the equation
$$\left(x - 3\right) \left(x + 1\right) - \frac{9}{4} = 0$$
We get the quadratic equation
$$x^{2} - 2 x - \frac{21}{4} = 0$$
This equation is of the form

a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -2$$
$$c = - \frac{21}{4}$$
, then

D = b^2 - 4 * a * c = 

(-2)^2 - 4 * (1) * (-21/4) = 25

Because D > 0, then the equation has two roots.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = \frac{7}{2}$$
$$x_{2} = - \frac{3}{2}$$