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

(A) To check if the first polynomial is a factor of the second,we perform long division of $3x^{4}+5x^{3}-7x^{2}+2x+2$ by $x^{2}+3x+1$.
Step $1$: Divide $3x^{4}$ by $x^{2}$ to get $3x^{2}$. Multiply $3x^{2}(x^{2}+3x+1) = 3x^{4}+9x^{3}+3x^{2}$. Subtracting this from the dividend gives $-4x^{3}-10x^{2}+2x+2$.
Step $2$: Divide $-4x^{3}$ by $x^{2}$ to get $-4x$. Multiply $-4x(x^{2}+3x+1) = -4x^{3}-12x^{2}-4x$. Subtracting this gives $2x^{2}+6x+2$.
Step $3$: Divide $2x^{2}$ by $x^{2}$ to get $2$. Multiply $2(x^{2}+3x+1) = 2x^{2}+6x+2$. Subtracting this gives a remainder of $0$.
Since the remainder is $0$,the first polynomial is a factor of the second polynomial.