$X$ 's

Dividing the whole equation by $x^2$ , we get $\color{#3D99F6}{x^2}-\color{#D61F06}{97x}+2012-\color{#D61F06}{\frac{97}{x}}+\color{#3D99F6}{\frac{1}{x^2}}=0$ $\small{\color{#EC7300}{Put~x+\frac{1}{x}=t~such~ that~x^2+\frac{1}{x^2}=t^2-2}}$ $\Rightarrow t^2-97t+2010=0$ $\Rightarrow (t-30)(t-67)=0$ $\Rightarrow t=x+\frac{1}{x}=30,67$ $\Rightarrow x^2-30x+1=0~and~x^2-67x+1=0$ Discriminant of both $x^2-30x+1=0~and~x^2-67x+1=0$ is greater than 0, hence will yield two solutions each hence a total of $\Large \color{#69047E}{\boxed{\color{#20A900}{4}}}$ solutions.

The degree of the equation is the number of solutions and also here nothing is said about complex or real roots so all answers are valid

Lol there is a high probability of getting this question right, because the greatest number of roots a fourth degree polynomial can have is 4, and the least is 0. So, the probability of getting this question right is $\frac{1}{3}$ .

May be, the highest power of variable = number of solutions.