For this let us consider an example of two trinomials: x3 + 4x + 8 and 5x3 + 2x2 + 7x + 1. Let us multiply them:
(x3 + 4x + 8) * (5x3 + 2x2 + 7x + 1),
We must pick one term at a time from any of the two polynomials and that should be multiplied to each term of another polynomial. This reduces complexity of problem. Only thing we need to be aware of is the sign of multiplication. Here, in our example we take terms of trinomial (x3 + 4x + 8) one by one and multiply them to each term in second trinomial (5x3 + 2x2 + 7x + 1) to get:
(x3 + 4x + 8) * (5x3 + 2x2 + 7x + 1) = x3 (5x3 + 2x2 + 7x + 1) + 4x (5x3 + 2x2 + 7x + 1) + 8 (5x3 + 2x2 + 7x + 1) = 5x6 + 2x5 + 7x4 + x3 + 20x4 + 8x3 + 28x2 + 4x + 40x3 + 16x2 + 56x + 8.
Reducing the polynomial by adding or subtracting the coefficients of terms with same degrees:
= 5x6 + 2x5 + 27x4 + 49x3 + 44x2 + 60x + 8,
We see that resultant polynomial is of higher degree.