To identify the types of roots

Question

If $(x+2)(x+3)(x+8)(x+12)=4x^{2}$ then the equation has what type of roots?

My attempt: Intersecting the graphs
Can you please suggest any other easy method?

Answer 1

Firstly, we can write $$(x+2)(x+3)(x+8)(x+12)-4x^2=(x+4)(x+6)(x^2+15x+24).$$

Also, we can write $$(x^2+14x+24)(x^2+11x+24)=4x^2$$ or $$\left(x+\frac{24}{x}+14\right)\left(x+\frac{24}{x}+11\right)=4$$ and after a substitution $x+\frac{24}{x}+11=t$ we can get all roots.

Indeed, we obtain $t^2+3t-4=0,$ which gives $t=1$ or $t=-4$,

For $x+\frac{24}{x}+11=1$ we obtain $\{-4,-6\}$ and the equation $x+\frac{24}{x}+11=-4$ has two real roots.