How to solve $x(3x+3)(x+5)(2x+12)+576 = 0$? 
How to solve $$x(3x+3)(x+5)(2x+12)+576 = 0?$$

This was a question on a test i recently took, And i wasn't able to solve it. I later tried to solve it using online calculators and it turns out this doesn't have any real solutions.
I know there's a general formula for quartic polynomials that can work but we were only taught two methods, modifying the equation to a quadratic one using substitution or guessing some of the solutions. Both of these didn't work for me.
Are there any ways to prove that this doesn't have real solutions without the quartic formula?

Another possible way is using the symmetry of $x(3x+3)(x+5)(2x+12) = 6x(x+1)(x+5)(x+6)$ around $x=\color{blue}{3}$:
$$6x(x+1)(x+5)(x+6) = 6(x+\color{blue}{3}-3)(x+\color{blue}{3}+3)(x+\color{blue}{3}-2)(x+\color{blue}{3}+2)$$ $$=6(\underbrace{(x+\color{blue}{3})^2}_{y:=}-9)((x+\color{blue}{3})^2-4)$$
The minimum value of $(y-9)(y-4)$ is $-\frac{25}{4}$. Hence,
$$6x(x+1)(x+5)(x+6)+576 \geq 6(-\frac{25}{4})+576 >0$$
