(x + a)(x + 2a)(x + 3a)(x + 4a) + a4
= (x + a)(x + 4a)(x + 2a)(x + 3a) + a4
= (x2 + 4ax + ax + 4a2)(x2 + 3ax + 2ax + 6a2) + a4
= (x2 + 5ax + 4a2)(x2 + 5ax + 6a2) + a4
Đặt x2 + 5ax + 4a2 = t
= t(t + 2a2) + a4
= (t + a2)2
= (x2 + 5ax + 4a2 + a2)2
= (x2 + 5ax + 5a2)2
Cách đặt khác ez hơn :))
\(A=\left(x+a\right)\left(x+2a\right)\left(x+3a\right)\left(x+4a\right)+a^4\)
\(A=\left[\left(x+a\right)\left(x+4a\right)\right]\left[\left(x+2a\right)\left(x+3a\right)\right]+a^4\)
\(A=\left(4a^2+5ax+x^2\right)\left(6a^2+5ax+x^2\right)+a^4\)
Đặt \(p=5a^2+5ax+x^2\)
\(\Rightarrow A=\left(p-a^2\right)\left(p+a^2\right)+a^4\)
\(\Rightarrow A=p^2-a^4+a^4\)
\(\Rightarrow A=p^2\)
Thay \(p=5a^2+5ax+x^2\)vào A ta có :
\(A=\left(5a^2+5ax+x^2\right)^2\)
