Let's construct a quadratic equation whose roots ($x_{1}$ and $x_{2}$) satisfy the following relationships:

$$
x_{1} x_{2} + x_{1} + x_{2} - a = 0, \quad x_{1} x_{2} - a(x_{1} + x_{2}) + 1 = 0.
$$
If the sought equation is of the form $x^{2}+p x+q=0$, then the given relationships can be written as:

$$
q-p=a \text { and } q+a p=-1
$$

from which: $p=-1, q=a-1$; thus, the sought equation is:

$$
x^{2}-x+a-1=0
$$

(Pál Füstös, Eger.)

Number of solutions: 46.