ANSWER :

Let q ( x ) = x^2 + 7x + 12

= x^2 + 4x + 3x + 12

= x ( x + 4 ) + 3 ( x + 4 )

= ( x + 4 )( x + 3 )

We know that

product of p( x ) and q ( x ) = product of H.C.F. and L.C.M.

p ( x ) × ( x + 3 )( x + 4 ) = ( x + 3 )( x + 4 )(x+ 3)^2

p ( x ) = ( x + 3 )^2

p ( x ) = x^2 + 6x + 9

THANKS !!!

Hello friends!!

Here is your answer :

The H.C.F and L.C.M. of polynomials p(x) and x² +7x + 12 are (x + 3) and (x+ 4)(x + 3)² respectively.The polynomials p(x) is

x² + 6x + 9

Solution:

Let g(x) = x² +7x + 12

We know that,

Product of p(x) and g(x) =

Product of p(x) and g(x) = H. C. F × L. C. M

First, we have to find the factors of g(x). Put g(x) = 0

x² +7x + 12 = 0

=> x² + ( 4 + 3 )x + 12 = 0

=> x² + 4x + 3x +12 = 0

=> x ( x + 4 ) + 3 ( x + 4 ) = 0

=> ( x + 4 ) ( x + 3 )

Putting the value in above formula :

▶️ p(x) ✖️ ( x + 4 ) ( x + 3 ) = ( x + 4 ) ( x + 3 )² × ( x + 3 )

▶️ p(x) × ( x + 4 ) ( x + 3 ) = ( x + 4 ) ( x + 3 )³

▶️ p(x)= ( x + 4 ) ( x + 3 )³ / ( x + 4 ) ( x + 3 )

▶️ p(x) = ( x + 3 )²

▶️ p(x) = (x)² + (3)² + 2 × 3 × x

▶️ p(x) = x² + 6x + 9

Hope it helps you