Question

1. $(p^5 - 12p^3 + 11p^2 - 16p + 16) div (p + 4)$

1. $(m^5 - 3m^4 + 4m^2 + 3m - 1) div (m + 1)$

Explanation:

For problem 5: $(p^5 - 12p^3 + 11p^2 - 16p + 16) \div (p+4)$

Step1: Set up polynomial long division

Arrange dividend with $0p^4$ term: $p^5 + 0p^4 -12p^3 +11p^2 -16p +16$, divisor $p+4$.

Step2: Divide leading terms

$\frac{p^5}{p}=p^4$. Multiply divisor by $p^4$: $p^5 +4p^4$. Subtract from dividend:
$(p^5 +0p^4 -12p^3) - (p^5 +4p^4) = -4p^4 -12p^3$

Step3: Divide new leading terms

$\frac{-4p^4}{p}=-4p^3$. Multiply divisor by $-4p^3$: $-4p^4 -16p^3$. Subtract:
$(-4p^4 -12p^3 +11p^2) - (-4p^4 -16p^3) = 4p^3 +11p^2$

Step4: Divide new leading terms

$\frac{4p^3}{p}=4p^2$. Multiply divisor by $4p^2$: $4p^3 +16p^2$. Subtract:
$(4p^3 +11p^2 -16p) - (4p^3 +16p^2) = -5p^2 -16p$

Step5: Divide new leading terms

$\frac{-5p^2}{p}=-5p$. Multiply divisor by $-5p$: $-5p^2 -20p$. Subtract:
$(-5p^2 -16p +16) - (-5p^2 -20p) = 4p +16$

Step6: Divide new leading terms

$\frac{4p}{p}=4$. Multiply divisor by $4$: $4p +16$. Subtract:
$(4p +16) - (4p +16) = 0$

---

For problem 7: $(m^5 - 3m^4 + 4m^2 + 3m - 1) \div (m+1)$

Step1: Set up polynomial long division

Arrange dividend with $0m^3$ term: $m^5 -3m^4 +0m^3 +4m^2 +3m -1$, divisor $m+1$.

Step2: Divide leading terms

$\frac{m^5}{m}=m^4$. Multiply divisor by $m^4$: $m^5 +m^4$. Subtract from dividend:
$(m^5 -3m^4 +0m^3) - (m^5 +m^4) = -4m^4 +0m^3$

Step3: Divide new leading terms

$\frac{-4m^4}{m}=-4m^3$. Multiply divisor by $-4m^3$: $-4m^4 -4m^3$. Subtract:
$(-4m^4 +0m^3 +4m^2) - (-4m^4 -4m^3) = 4m^3 +4m^2$

Step4: Divide new leading terms

$\frac{4m^3}{m}=4m^2$. Multiply divisor by $4m^2$: $4m^3 +4m^2$. Subtract:
$(4m^3 +4m^2 +3m) - (4m^3 +4m^2) = 0m^2 +3m$

Step5: Divide new leading terms

$\frac{3m}{m}=3$. Multiply divisor by $3$: $3m +3$. Subtract:
$(3m -1) - (3m +3) = -4$

Answer:

1. $p^4 - 4p^3 + 4p^2 - 5p + 4$
2. $m^4 - 4m^3 + 4m^2 + 3 - \frac{4}{m+1}$