$\\(x+2)(x+4)(x+6)(x+8)+2015\\=(x^2+8x+2x+16)(x^2+6x +4x+24)+2015\\=(x^2+10x+16)(x^2+10x+24)+2015$

Đặt $x^2+10x+16=t$

$=t(t+8)+2015\\=t^2+8t+16 + 1999\\=(t+4)^2+1999\\=(x^2+10x+16+4)^2+1999\\=(x^2+10x+20)^2-1^2+2000\\=(x^2+10x+20-1)(x^2+10x+20+1)+2000\\=(x^2+10x+19)(x^2+10x+21)+2000$

Do $(x^2+10x+21)\vdots (x^2+10x+21)$

$\Rightarrow (x^2+10x+21)(x^2+10x+19)\vdots (x^2+10x+21)$

$\Rightarrow (x^2+10x+21)(x^2+10x+19)+2000\vdots (x^2+10x+21)$ (Dư $2000$)

$\Rightarrow (x+2)(x+4)(x+6)(x+8)+2015$ chia $x^2+10x+21$ dư $2000$

Vậy $(x+2)(x+4)(x+6)(x+8)+2015$ chia $x^2+10x+21$ dư $2000$

Đáp án + giải thích các bước giải:

`(x + 2)(x + 4)(x + 6)(x + 8) + 2015`

` = [(x + 2)(x + 8)][(x + 4)(x + 6)] + 2015`

` = (x^2 + 2x + 8x + 16)(x^2 + 4x + 6x + 24) + 2015`

` = (x^2 + 10x + 16)(x^2 + 10x + 24) + 2015`

Đặt `t = x^2 + 10x + 20`

`=> (t - 4)(t + 4) + 2015`

` = t^2 - 16 + 2015`

` = t^2 + 1999`

Mà `t + 1 = x^2 + 10x + 21`

`=> t^2 + 1999 : (t + 1)`

Vậy `(x + 2)(x + 4)(x + 6)(x + 8) + 2015` chia cho `x^2 + 10x +  21` dư `2000`