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

(N/A) To check if the first polynomial is a factor of the second,we perform polynomial long division.
Dividing $x^{5}-4x^{3}+x^{2}+3x+1$ by $x^{3}-3x+1$:
$1$. Divide the first term of the dividend $(x^{5})$ by the first term of the divisor $(x^{3})$ to get $x^{2}$.
$2$. Multiply $x^{2}$ by $(x^{3}-3x+1)$ to get $x^{5}-3x^{3}+x^{2}$.
$3$. Subtract this from the dividend: $(x^{5}-4x^{3}+x^{2}+3x+1) - (x^{5}-3x^{3}+x^{2}) = -x^{3}+3x+1$.
$4$. Divide the first term of the new polynomial $(-x^{3})$ by the first term of the divisor $(x^{3})$ to get $-1$.
$5$. Multiply $-1$ by $(x^{3}-3x+1)$ to get $-x^{3}+3x-1$.
$6$. Subtract this from the current remainder: $(-x^{3}+3x+1) - (-x^{3}+3x-1) = 2$.
Since the remainder is $2$ (which is $\neq 0$),the first polynomial is not a factor of the second polynomial.