Classify the following polynomial as a polynomial in one variable,two variables,etc.:
$x^{2}-2xy+y^{2}+1$

Divide $p(x) = x^{3} + 7x^{2} + 14x + 1$ by $x + 3$ and find the quotient and the remainder.

Which of the following expressions is a polynomial,and state the reason? If an expression is a polynomial,state whether it is a polynomial in one variable or not: $x^{3}+3 x^{2} y+3 x y^{2}+y^{3}$

On dividing $p(x) = 2x^3 - 3x^2 + ax - 3a + 9$ by $(x + 1)$,if the remainder is $16$,then find the value of $a$. Then,find the remainder on dividing $p(x)$ by $(x + 2)$.

On dividing $p(x) = x^{3} + 2x^{2} - 5ax - 7$ by $(x + 1)$,the remainder is $R_{1}$ and on dividing $q(x) = x^{3} + ax^{2} - 12x + 6$ by $(x - 2)$,the remainder is $R_{2}$. If $2R_{1} + R_{2} = 6$,then find the value of $a$.

One of the factors of $(25 x^{2}-1)+(1+5 x)^{2}$ is