i) factorize 8p²+2p-8pq-2q ii) factorize 8x²+32xy+32y² iii) using results of i) and ii), factorize (8x²+32xy+32y²)+2x+4y-8(x+2y)(y-3)-2y+6?

Sol
i)
8p^2+2p－8pq－2q
=(8p^2－2p)－(8pq+2q)
=2p(4p－1)－2q(4p+1)
=(2p－2q)(4p+1)
=2(p－q)(4p+1)
ii)
8x^2+32xy+32y^2
=8(x^2+4xy+4y^2)
=8(x+2y)^2
iii)
Set p=x+2y，q=y－3
p－q=(x+2y)－(y－3)=x+y+3
(8x^2+32xy+32y^2)+2x+4y－8(x+2y)(y－3)－2y+6
=8(x+2y)^2+2(x+2y)－8(x+2y)(y－3)－2(y－3)
=8p^2+2p－8pq－2q
=2(p－q)(4p+1)
=2(x+y+3)(4x+8y+1)

1.(8p+2)(p-q)
2.(4x+6y)(2x+6y)
=2(2x+3y)(x+3y)
3.2(x+2y)(x+3y)+2(x+2y)-8(x+2y)(y-3)-2(y-3)
=2(x+2y)(x+3y+1)-2(y-3)(4x-8y+1)
=4(x+2y)(y+3)(x+3y+1)(4x+8y-1)