When factoring a trinomial, why is it necessary to write the trinomials in four terms?"

You keep posting this without any thinking.

Can this be factored in four terms?
10x^5 + 2x + 17

Perhaps you are meaning factoring the "quadratic" (a polynomial of degree 2) equation. Often a quadratic equation is a trinomial. The reason for that is that the degree 2 solution always involves two solutions, so...
(x-a)(x-b)=0 is the only way to get two solutions.

I still don't understand?
Yes I have been thinking alot on this one...i am just not understanding it

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