Find all polynomials that satisfy a functional equation

Question

I need help with this problem if anyone can contribute to solving this I would much appreciate it:

Find all polynomials $P$ that satisfy $P(1)=210$ and for every $x\in \mathbb{R}$: $$(x+10)P(2x)=(8x-32)P(x+6).$$

Answer 1

$$P(x) = c (-8 + x) (-4 + x) (4 + x)$$

Answer 2

small hint

With $ x=-10 $, we get $$P(-10+6)=P(-4)=0$$ and for $ x=4 $, it gives $$P(8)=0$$

thus

$$P(x)=(x-8)(x+4)R(x)$$ with $$R(1)=\frac{210}{-35}=-6$$