Find the square root of following polynomial $(6 x^{2} + 5 x - 6) (6 x^{2} - x - 2) (4 x^{2} + 8 x + 3)$

| (2 x + 3) (3 x - 2) (2 x + 1) |

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| (2 x + 3) (3 x + 2) (2 x + 1) |

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| (2 x - 3) (3 x - 2) (2 x + 1) |

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None of these

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Solution

The correct option is A $| (2 x + 3) (3 x - 2) (2 x + 1) |$
We find the square root of $(6 x^{2} + 5 x - 6) (6 x^{2} - x - 2) (4 x^{2} + 8 x + 3)$ as shown below:

$= \sqrt{(6 x^{2} + 5 x - 6) (6 x^{2} - x - 2) (4 x^{2} + 8 x + 3)}$

$= \sqrt{(6 x^{2} + 9 x - 4 x - 6) (6 x^{2} + 3 x - 4 x - 2) (4 x^{2} + 2 x + 6 x + 3)}$

$= \sqrt{[3 x (2 x + 3) - 2 (2 x + 3)] [3 x (2 x + 1) - 2 (2 x + 1)] [2 x (2 x + 1) + 3 (2 x + 1)]}$

$= \sqrt{[(3 x - 2) (2 x + 3)] [(3 x - 2) (2 x + 1)] [(2 x + 3) (2 x + 1)]}$

$= \sqrt{(3 x - 2) (2 x + 3) (3 x - 2) (2 x + 1) (2 x + 3) (2 x + 1)}$

$= \sqrt{(3 x - 2)^{2} (2 x + 3)^{2} (2 x + 1)^{2}}$

$= \sqrt{[(3 x - 2) (2 x + 3) (2 x + 1)]^{2}}$

$= (2 x + 3) (3 x - 2) (2 x + 1)$

Hence, the square root of $(6 x^{2} + 5 x - 6) (6 x^{2} - x - 2) (4 x^{2} + 8 x + 3)$ is $(2 x + 3) (3 x - 2) (2 x + 1)$ .

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