One example is adding x^3+x^2 with 7x^2-9x which are both binomials since they have two terms each.
The like terms are the x^2 and 7x^2 which combine to 8x^2. The other terms stay as they are.
So overall, (x^3+x^2) + (7x^2-9x) = x^3 + 8x^2 - 9x which is a trinomial as it has three terms
-----------------------------------------------------------------------------------------
Another example is adding x^2 - 4 with 2x+7 to get x^2 + 2x + 3