Using a suitable identity,evaluate the following:
$101 \times 102$

Determine whether $(x+1)$ is a factor of the polynomial $p(x) = x^{3} - 2x^{2} - 5x + 6$.

For the polynomial $p(x) = x^{2} - 7x + 12$,find the value of $p(2)$.

Find the zeroes of the polynomial:
$p(x)=(x-2)^{2}-(x+2)^{2}$

The polynomial $p(x) = x^{4} - 2x^{3} + 3x^{2} - ax + 3a - 7$ when divided by $x + 1$ leaves the remainder $19$. Find the value of $a$. Also,find the remainder when $p(x)$ is divided by $x + 2$.

If the polynomials $ax^3 + 4x^2 + 3x - 4$ and $x^3 - 4x + a$ leave the same remainder when each is divided by $x - 3$,find the value of $a$.