Question

1. $(n^3 + 2n^2 - 41n - 48) div (n + 7)$

Explanation:

Step1: Divide leading terms

$\frac{n^3}{n} = n^2$
Multiply divisor by $n^2$: $(n+7)n^2 = n^3 +7n^2$
Subtract from dividend:
$(n^3+2n^2-41n-48)-(n^3+7n^2) = -5n^2-41n-48$

Step2: Divide new leading terms

$\frac{-5n^2}{n} = -5n$
Multiply divisor by $-5n$: $(n+7)(-5n) = -5n^2-35n$
Subtract from current polynomial:
$(-5n^2-41n-48)-(-5n^2-35n) = -6n-48$

Step3: Divide new leading terms

$\frac{-6n}{n} = -6$
Multiply divisor by $-6$: $(n+7)(-6) = -6n-42$
Subtract from current polynomial:
$(-6n-48)-(-6n-42) = -6$

Answer:

$n^2 - 5n - 6 - \frac{6}{n+7}$