Factorize 2x⁴ + x³ – 14x² – 19x – 6 by x² + 3x + 2 and verify the division algorithm.

We stop the division process here, since remainder r(x) = 0
Here,
Divisor g(x) =

straight x squared plus 3 straight x plus 2

Dividend p(x) =

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Quotient q(x) =

2 straight x squared minus 5 straight x minus 3

and Remainder r(x) = 0
Now,

Divisor space cross times space Quotient space plus space Remainder

space space equals space left parenthesis straight x squared plus 3 straight x plus 2 right parenthesis space left parenthesis 2 straight x squared minus 5 straight x minus 3 right parenthesis plus 0

equals 2 straight x to the power of 4 minus 5 straight x cubed minus 3 straight x squared plus 6 straight x cubed minus 15 straight x squared minus 9 straight x plus 4 straight x squared minus 10 straight x minus 6

equals 2 straight x to the power of 4 plus straight x cubed minus 14 straight x squared minus 19 straight x minus 6

= Dividend
Therefore, the division algorithm verified.

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Factorize 2x4 + x3 – 14x2 – 19x – 6 by x2 + 3x + 2 and verify the division algorithm.

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Factorize 2x⁴ + x³ – 14x² – 19x – 6 by x² + 3x + 2 and verify the division algorithm.