Prove that $(x^3 - x)(2x^2 + 5x - 3)$ is divisible by $5$ without using induction

Question

I've factorised the equation to $x(x+1)(x-1)(2x-1)(x+3)$ and I notice it has $3$ consecutive numbers, a missing one and then the next consecutive number not sure how to use this to prove the required divisibility.

Answer 1

as you says, we have $$(x-1)x(x+1)(x+3) (2x-1) ~,$$ so we just need to proof that: when $(x-1)x(x+1)(x+3) \not \equiv 0\pmod 5$, $2x-1 \equiv 0 \pmod 5$.

$(x-1)x(x+1)(x+3) \not \equiv 0\pmod 5$ means that $x+2 \equiv 0 \pmod 5$ so $x \equiv 3 \pmod 5$.then $2x-1 \equiv 6-1 \equiv 0 \pmod 5$.

Q.E.D