the polynomial in a, a+1, is not a factor of the polynomial in a, a^2 + 1.

this does not mean that for some particular NUMBER a, a+1 is not a factor of a^2 + 1, which is clearly false: for example a = 1 gives 2 divides 2, which is true.

there is a difference between the function defined by f(a) = a+1, and the value f(a) for some particular a.

two unequal functions can nevertheless have intersecting graphs.