How many roots does the polynomial $acx^4 + b(a + c)x^3 + (a^2 + b^2 + c^2)x^2 + b(a + c)x + ac$ have

Question

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

I have no clue as to how to approach it, a hint would suffice

Answer 1

HINT: write your equation in the form $$(x(ax+b)+c)(a+x(b+cx))=0$$ and solve it

Answer 2

HINT: use Descartes' rule of signs and check conditions to get a certain number of roots of a certain type...