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A counter-example proves the falsity:

Consider the two binomials (x + 2) and (x - 2). Then

(x + 2)(x - 2) = x2 - 2x + 2x - 4

= x2 - 4

another binomial.

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Q: Will the product of two binomials after combining like terms always be trinomial?
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Will the product of two binomials always equal a trinomial?

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The product of a number and its reciprocal is always one. That's what reciprocal means.

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That is false. The product of two negative integers is always positive.

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