第1章 1.29

(1)

$(x+y+3)(x+y-2)=(x+y)^2+(x+y)-6=x^2+2xy+y^2+x+y-6$

(2)

$(2x+y+1)(2x+y-3)=(2x+y)^2-2(2x+y)-3=4x^2+4xy+y^2-4x-2y-3$

(3)

$(x-2y-3)(x-2y+2)=(x-2y)^2-(x-2y)-6=x^2-4xy-4y^2-x+2y-6$

(4)

$(x^2+x+1)(x^2-x+1)=(x^2+1)^2-x^2=x^4+2x^2+1-x^2=x^4+x^2+1$

(5)

$(x+1)(x+2)(x+3)(x+4)=(x^2+3x+2)(x^2+7x+12)$

$=x^4+7x^3+12x^2+3x^3+21x^2+36x+2x^2+14x+24=x^4+10x^3+35x^2+50x+24$

(6)

$(x-1)(x+1)(x^4+x^2+1)=(x^2-1)(x^4+x^2+1)$

$=x^6+x^4+x^2-x^4-x^2-1=x^6-1$

(7)

$(a+b-c)^2=(a+b)^2-2c(a+b)-c^2=a^2+2ab+b^2-2ac-2bc+c^2$

(8)

$(x-2y+3)^2=(x-2y)^2+6(x-2y)+9=x^2-4xy-4y^2+6x-12y+9$

(9)

$(a+b+c)(a+b-c)(a-b+c)(a-b-c)=\{(a+b)^2-c^2\}\{(a-b)^2-c^2\}$

$=(a^2+b^2-c^2+2ab)(a^2+b^2-c^2-2ab)$

$=(a^2+b^2-c^2)^2-4a^2b^2$

$=a^4+b^4+c^4-2a^2b^2-2b^2c^2-2a^2c^2$