Let a, b and c be real numbers. Then the fourth degree polynomial in $x$, $acx^4 + b(a + c)x^3 + (a^2 + b^2 + c^2 )x^2 + b(a + c)x + ac$
(a) Has four complex (non-real) roots
(b) Has either four real roots or four complex roots
(c)Has two real roots and two complex roots
(d) Has four real roots
This question is from a book called 'Test of Mathematics at the $10+2$ Level' published by the Indian Statistical Institute
We can see that the expression factorizes to $(ax^2+bx+c)(a+bx+cx^2)$. If $\alpha,\beta$ are the roots of the first factor then $1/\alpha , 1/\beta$ are the roots of the second expression. And we know that complex roots occur in conjugation. So, we can easily understand that option (b) is a correct.
But today, I want to solve this problem in some different way in which it does not require us to factorize the expression in a hit and trial way as I did. Even if we have to factorize, we must use a proper way to find the factors. Please help.