How to solve this in an efficient way (without calculators) 
Solve for $x$, if
  $$(x+4)(x+7)(x+8)(x+11)+20=0.$$

Is there an easier way to solve this than trying to multiply all the values together? I've tried multiplying all of them together, and I get an equation of the fourth degree which I find very hard to factorise. I'm hoping there is an easier way to solve this question. Any help is appreciated.Thanks :).
The equation I get after multiplying is
$$x^4+30x^3+325x^2+1500x+2484=0,$$
and its roots are 
$$-6, \quad -9, \quad \frac{-15\pm\sqrt{41}}2 .$$

$f(x) = (x+4)(x+7)(x+8)(x+11)+20\ $ is symmetric about the line $x = -7.5$
$y = x + 7.5$
$(y-3.5)(y-0.5)(y+0.5)(y+3.5)+20=0$ 
$(y^2 - 0.5^2)(y^2 -3.5^2)+20=0$ 
$y^4 - 12.5 y^2 + 23.0625 = 0$
Use the quadratic formula to solve for $y^2.$
The square roots of that give $y.$
And, subtract $7.5$ to get $x.$
