分解因式:4(3x²-x-1)(x²+2x-3)-(4x²+x-4)²

分解因式:4(3x²-x-1)(x²+2x-3)-(4x²+x-4)²
分解因式:4(3x²-x-1)(x²+2x-3)-(4x²+x-4)²

分解因式:4(3x²-x-1)(x²+2x-3)-(4x²+x-4)²
看一看
( 3x" - x - 1 ) + ( x" + 2x - 3 ) = 4x" + x - 4
发现了什么吗?
4( 3x" - x - 1 )( x" + 2x - 3 ) - ( 4x" + x - 4 )"
题目想到了怎么办吗?
= 4[ ( 4x" + x - 4 ) - ( x" + 2x - 3 ) ]( x" + 2x - 3 ) - ( 4x" + x - 4 )"
= 4( 4x" + x - 4 )( x" + 2x - 3 ) - 4( x" + 2x - 3 )" - ( 4x" + x - 4 )"
= - [ ( 4x" + x - 4 )" - 4( 4x" + x - 4 )( x" + 2x - 3 ) + 4( x" + 2x - 3 )" ]
这就像 12" =144,也是完全平方啊
= - [ ( 4x" + x - 4 ) - 2( x" + 2x - 3 ) ]"
= - ( 4x" + x - 4 - 2x" - 4x + 6 )"
= - ( 2x" - 3x + 2 )"
或者
= 4( 3x" - x - 1 )[ ( 4x" + x - 4 ) - ( 3x" - x - 1 ) ] - ( 4x" + x - 4 )"
= 4( 3x" - x - 1 )( 4x" + x - 4 ) - 4( 3x" - x - 1 )" - ( 4x" + x - 4 )"
= - [ 4( 3x" - x - 1 )" - 4( 3x" - x - 1 )( 4x" + x - 4 ) + ( 4x" + x - 4 )" ]
这就像 21" =441,也是完全平方式
= - [ 2( 3x" - x - 1 ) - ( 4x" + x - 4 ) ]"
= - ( 6x" - 2x - 2 - 4x" - x + 4 )"
= - ( 2x" - 3x + 2 )"

令t=4x²+x-4
原式=4[t-(x²+2x-3)](x²+2x-3)-t²
=-t²+4t(x²+2x-3)-4(x²+2x-3)²
=-[t²-4t(x²+2x-3)+4(x²+2x-3)²]
=-[t-2(x²+2x-3)]²
=-(4x²+x-4-2x²-4x+6)²
=-(2x²-3x+2)²