Example 1. For what integer values of $x$ is $f(x)=x^{2}+5 x-20$ a perfect square?
Solution: Since $\Delta=5^{2}-4 \times(-20)=3 \times 5 \times$ 7, $\Delta$ has the following eight decompositions $p_{2} p_{1}\left(p_{1} \geqslant p_{2}\right): 7$ $\times 15,5 \times 21,3 \times 35,1 \times 105,(-7) \times(-15)$, $(-5) \times(-21),(-3) \times(-35),(-1) \times$ (-105). According to the conclusion 1 of the theorem, when $x$ takes the values $3,4,7,24,-8,-9,-12,-29$, the value of $f(x)$ is a perfect square.