Estonian Mathematical Olympiad

Using equalities $2025=45^2$ and $90=2\cdot 45$ we get $$\begin{array}{rl}{x}^4-2x^3-88x^2+90x+2025& =x^4-2x^3-89x^2+(x+45{)}^2\\ & =(x^2{)}^2-2x^2(x+45)+(x+45{)}^2+x^2\\ & =(x^2-x-45{)}^2+x^2.\end{array}$$ But $(x^2-x-45{)}^2+x^2\ge 0$. Equality is impossible because it could occur only if $x=0$ and also $x^2-x-45=0$. Thus indeed $(x^2-x-45{)}^2+x^2>0$.