Using the method of undetermined coefficients. Show that the expression $(x + 1) (x + 2) (x + 3) (x + 4) + 1$ is the square of a trinomial.

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Solution

$(x + 1) (x + 2) (x + 3) (x + 4) + 1$

$= (x + 1) (x + 4) (x + 2) (x + 3) + 1$

$= (x^{2} + 5 x + 4) (x^{2} + 5 x + 6) + 1$

Let, $x^{2} + 5 x + 4 = K$ then

$= K (K + 2) + 1$

$= K^{2} + 2 K + 1$

$= (K + 1)^{2}$

$= (x^{2} + 5 x + 4 + 1)^{2}$

$= (x^{2} + 5 x + 5)^{2}$

Hence proved.

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