The expression $(x + 1) (x + 2) (x + 3) (x + 4) + 1$ is a

perfect square

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cube

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quartic polynomial

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none of the above

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Solution

The correct option is A perfect square
Solving
$(x + 1) (x + 2) (x + 3) (x + 4) + 1$
$M u l t i p l y i n g f i r s t b r a c k e t w i t h l a s t a n d s e c o n d t o t h e t h i r d o n e$
$(x^{2} + 5 x + 4) (x^{2} + 5 x + 6) + 1$
$R e p l a c i n g x^{2} + 5 x + 4 b y^{'} B^{'}$
$(B) (B + 2) + 1$
$B^{2} + 2 B + 1$
$(B + 1)^{2} = (x^{2} + 5 x + 5)^{2}$
Hence $L . H . S .$ is the $P e r f e c t S q u a r e$ of $(x^{2} + 5 x + 5)$

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