$\displaystyle a+1$ is not a factor of $\displaystyle a^2+1$.
However, when $\displaystyle a=-3$, we get -2 is not a factor of 10, which is false. Could someone explain this anomaly?
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.
Because it's not true for all the values you'll give to a, take for example a=5 then a+1=6 which is not a factor of a^2+1=26. If you take for example a^2-1, then a-1 is a factor, so for every value of a it will still be a factor.