Question

Factorize following expressions
0
(x + 1)(x+2) (x+3)(x + 4) - 15​

Answer 1

Answer:

We know that  a2−b2=(a+b)(a−b) , so it would be nice if we could write  (x+1)(x+2)(x+3)(x+4)  as a square. We can’t see how to do that, but because of the symmetry we can try to express it as the product of two similar terms:

(x+1)(x+2)(x+3)(x+4)=(x+1)(x+4)(x+2)(x+3)=(x2+5x+4)(x2+5x+6) .

Thus, our expression is  (x2+5x+4)(x2+5x+6)−15 , which has ‘last coefficient’  9 . Now look at the  4  and the  6;  they sum to  10 . If we replace them with two other numbers summing to  10 , say  s  and  t , only the last coefficient will change (to  st ). So we seek  s  and  t  such that  s+t=10 , and  st=9 . This is clearly satisfied by  s=1  and  t=9 . Hence:

(x+1)(x+2)(x+3)(x+4)−15=(x2+5x+1)(x2+5x+9) , as required.

Step-by-step explanation: