One of the factors of $p(x) = x^{3} + 2x^{2} + 3x + 2$ is:

What must be subtracted from $8x^4 + 14x^3 - 2x^2 + 8x - 12$ so that the resulting polynomial is exactly divisible by $4x^2 + 3x - 2$?

Answer the following and justify: If on division of a non-zero polynomial $p(x)$ by a polynomial $g(x)$,the remainder is zero,what is the relation between the degrees of $p(x)$ and $g(x)$?

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
$x^{2}+3x+1, 3x^{4}+5x^{3}-7x^{2}+2x+2$

Answer the following and justify: Can $x^{2}-1$ be the quotient on division of $x^{6}+2 x^{3}+x-1$ by a polynomial in $x$ of degree $5$?

Obtain the cubic polynomial in the standard form with the following coefficients: $a=2, b=5, c=1, d=2$.