# 1. CONDITION

Consider quadratic trinomials of the form $x^{2}+p x+q$ with integer coefficients, where $p+q=30$. How many such trinomials have integer roots?
Solution. Let $x_{1}$ and $x_{2}$ be the integer roots of a quadratic polynomial of the form $x^{2} + px + q$. Then $p = -(x_{1} + x_{2})$, $q = x_{1} x_{2}$. Therefore, $30 = p + q = (x_{1} - 1)(x_{2} - 1) - 1$, i.e., $(x_{1} - 1)(x_{2} - 1) = 31$. Since the number 31 is prime, it can be represented as the product of two integers in only the following ways: $31 = 1 \cdot 31 = (-1) \cdot (-31)$. In the first case, the roots of the quadratic polynomial are the numbers 2 and 32 ($x^{2} - 34x + 64$), and in the second case, the roots are 0 and -30 (the quadratic polynomial $x^{2} + 30x$).

Answer. Two quadratic polynomials: $x^{2} - 34x + 64$ and $x^{2} + 30x$.