Given the quadratic equation $x^{2}+p x+q$ with roots $x_{1}$ and $x_{2}$. Construct the quadratic equation whose roots are $x_{1}+1$ and $x_{2}+1$.
Let $x^{2}+r x+s=0$ be the sought equation. Then

$$
x_{1}+1+x_{2}+1=-r
$$

and

$$
\left(x_{1}+1\right)\left(x_{2}+1\right)=x_{1} x_{2}+x_{1}+x_{2}+1=s
$$

But according to the problem,

$$
x_{1}+x_{2}=-p
$$

and

$$
x_{1} x_{2}=q
$$

thus

$$
r=p-2, \quad s=q-p+1
$$

and hence

$$
x^{2}+(p-2) x+q-p+1=0
$$

is the sought equation.

(Béla Koffer, Budapest.)

The problem was also solved by: Bendl K., Chambrée M., Csada I., Dénes M., Erdős V., Fried E., Füstös P., Kirchknopf E., Kiss J., Kovács Gy., Lázár B., Miklóssy K., Neumann L., Paunz A., Sárközy P., Strasser I., Szilárd V., Tandlich E., Viola R., Wáhl V.