Which of the following is/are correct regarding solution to the equation $(x+5)(x+6)(x+7)=504$?

(A) One real solution
(B) Two complex solutions
(C) Three real solutions
(D) No real solution
Solution

Let, $x+6 = y$

So, the equation becomes $(y-1).y.(y+1)=504$

$\Rightarrow (y^2-1)y=504$

$\Rightarrow y^3-y-504=0$

By hit and trial it can be seen that $y=8$ satisfies the above equation.

So, $y^2 (y-8) + 8y (y-8) + 63 (y-8)=0$

$\Rightarrow (y-8)(y^2+8y+63)=0$

$\therefore y=8,-4\pm \sqrt {47} i=x+6$

$\therefore x=2,-10\pm \sqrt {47} i$

Hence, (A) & (B)