Divide: $4 x^{5} + 7 x^{4} + 14 x^{3} + 3 x^{2} - 8 x + 6$ by $x^{2} + 3 x + 2$

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Solution

Steps:
1) Multiply $(x^{2} + 3 x + 2)$ with $4 x^{3}$ , to get $4 x^{5}$ , remainder $= - 5 x^{4} + 6 x^{3} + 3 x^{2}$
2) Multiply $(x^{2} + 3 x + 2)$ with $- 5 x^{2}$ , to get $- 5 x^{4}$ , remainder $= 21 x^{3} + 13 x^{2} - 8 x$
3) Multiply $(x^{2} + 3 x + 2)$ with $21 x$ to get, $21 x^{3}$ , Remainder $= - 50 x^{2} - 50 x + 6$
4) Multiply $(x^{2} + 3 x + 2)$ with $- 50$ , to get $- 50 x^{2}$ , remainder $= 100 x + 106$
Clearly, degree $(100 x + 106) = 1 <$ degree $(x^{2} + 3 x + 2) = 2$
$∴$ Division cannot be proceeded further,

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