Question

What is to be added to (x+a)(x+3a)(x+5a)(x+7a) so as to make it a perfect square

Answer 1

as you can observe that
(x+a)(x+7a) = $(x^{2} + 8ax + 7 a^{2})$
and
(x+3a)(x+5a) = $(x^{2} + 8ax + 15 a^{2})$

If we can make the $a^{2}$ terms equal, it will be perfect square.

(1) subtract $(x^{2} + 8ax + 7 a^{2})(8 a^{2})$
(2) add $(8 a^{2})(x^{2} + 8ax + 15 a^{2})$

(1) Subtract
$(x^{2} + 8ax + 7 a^{2})(x^{2} + 8ax + 15 a^{2})$ - $(x^{2} + 8ax +7a^{2})(8 a^{2})$
-> $(x^{2} + 8ax + 7 a^{2})(x^{2} + 8ax + 15 a^{2} - 8 a^{2})$
-> $(x^{2} + 8ax + 7 a^{2})(x^{2} + 8ax + 7 a^{2})$

(2) Add
$(x^{2} + 8ax + 7 a^{2})(x^{2} + 8ax + 15a^{2})$ + $(x^{2} + 8ax +15 a^{2})(8 a^{2})$
-> $(x^{2} + 8ax + 15 a^{2})(x^{2} + 8ax + 7 a^{2} + 8 a^{2})$
-> $(x^{2} + 8ax + 15 a^{2})(x^{2} + 8ax + 15 a^{2})$

Answer 2

(x+a)(x+7a)(x+3a)(x+5a)
(x²+8ax+7a²)(x²+8ax+15a²)
now if they are to be perfect square then we must make each expression a sqaure by themselves so we add and subtract by the opposite 
(x²+8ax+7a²)(x²+8ax+15a²)-(x²+8ax+7a²)8a²
taking common we have
(x²+8ax+7a²)(x²+8ax+15a²-8a²)
(x²+8ax+7a²)(x²+8ax+7a²)= (x²+8ax+7a²)²
similarly we have
(x²+8ax+7a²)(x²_8ax+15a²)+(x²+8ax+15a²)8a²
(x²+8ax+15a²)(x²+8ax+7a²+8a²)= (x²+8ax+15a²)²