To multiply two polynomials, multiply every term of the first by
every term of the second, then collect like terms. For two binomials
such as (2x + 5)(5x - 6) this gives four products —
often remembered as FOIL (First, Outer, Inner, Last):
(2x + 5)(5x - 6) = \underbrace{10x^2}_{\text{First}} \underbrace{- 12x}_{\text{Outer}} + \underbrace{25x}_{\text{Inner}} \underbrace{- 30}_{\text{Last}}
Then combine the two middle terms (-12x + 25x = 13x):
(2x + 5)(5x - 6) = 10x^2 + 13x - 30
The result is a quadratic: the highest power is
x^2, because an x from
each factor multiplied together gives x \cdot x = x^2.