Without actual division prove that $X^{4} + 2 X^{3} + 2 X^{2} + 2 X - 3$ is exactly divisible by $X^{2} + 2 X - 3$ .

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Solution

Proof.
Factorize the polynomial $X^{4} + 2 X^{3} + 2 X^{2} + 2 X - 3$ .
$X^{4} + 2 X^{3} + 2 X^{2} + 2 X - 3 = X^{4} + 2 X^{3} - 3 X^{2} + X^{2} + 2 X - 3 = X^{2} (X^{2} + 2 X - 3) + 1 (X^{2} + 2 X - 3) = (X^{2} + 1) (X^{2} + 2 X - 3)$
Since, $X^{2} + 2 X - 3$ is a factor of $X^{4} + 2 X^{3} + 2 X^{2} + 2 X - 3$ .
Hence, $X^{4} + 2 X^{3} + 2 X^{2} + 2 X - 3$ is exactly divisible by $X^{2} + 2 X - 3$ .

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