Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder in the following case:
$p(x) = x^{4} - 3x^{2} + 4x + 5$,$g(x) = x^{2} + 1 - x$

(N/A) To divide $p(x)$ by $g(x)$,we first write them in standard form (descending order of powers):
$p(x) = x^{4} + 0x^{3} - 3x^{2} + 4x + 5$
$g(x) = x^{2} - x + 1$
Performing polynomial long division:
$1$. Divide the first term of $p(x)$ by the first term of $g(x)$: $x^{4} / x^{2} = x^{2}$. This is the first term of the quotient.
$2$. Multiply $x^{2}$ by $(x^{2} - x + 1) = x^{4} - x^{3} + x^{2}$. Subtract this from $p(x)$ to get $x^{3} - 4x^{2} + 4x + 5$.
$3$. Divide the first term of the new polynomial by the first term of $g(x)$: $x^{3} / x^{2} = x$. This is the second term of the quotient.
$4$. Multiply $x$ by $(x^{2} - x + 1) = x^{3} - x^{2} + x$. Subtract this to get $-3x^{2} + 3x + 5$.
$5$. Divide the first term of the new polynomial by the first term of $g(x)$: $-3x^{2} / x^{2} = -3$. This is the third term of the quotient.
$6$. Multiply $-3$ by $(x^{2} - x + 1) = -3x^{2} + 3x - 3$. Subtract this to get $8$.
Thus,the quotient is $x^{2} + x - 3$ and the remainder is $8$.