<s>[INST] The number of real roots of $x^5 + 2x^3 + x^2 + 2 = 0 $ is The number of real roots of $x^5 + 2x^3 + x^2 + 2 = 0 $ is   
A. 0;
B. 3;
C. 5;
D. 1.
I don't know how to solve this.
 [/INST] An idea following the comment to your question:
$$x^5+2x^3+x^2+2=x^3(x^2+2)+x^2+2=(x^3+1)(x^2+2)$$
Thus, the only real roots can come from $\;x^3+1\;$, and this has only one since
$$x^3+1=(x+1)(x^2-x+1)$$
and the above quadratic factor is irreducible.
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