(x+1)(x+2)(x+3)(x+4)+10分解因式

来源：学生作业帮编辑：作业帮分类：数学作业时间：2024/05/23 23:01:16

(x + 1)(x + 2)(x + 3)(x + 4) + 10
= ( x + 1 )( x + 4 )( x + 3 )( x + 2 ) + 1 + 9
= ( x" + 4x + x + 4 )( x" + 2x + 3x + 6 ) + 1 + 9
= ( x" + 5x + 5 - 1 )( x" + 5x + 5 + 1 ) + 1 + 9
= ( x" + 5x + 5 )" - 1" + 1" + 9
= ( x" + 5x + 5 )" + 9
——要知道,连续4个数字的乘积再加1,也肯定是一个平方数
= ( x" + 5x + 5 )" - 5" + 25 + 9
= ( x" + 5x + 5 - 5 )( x" + 5x + 5 + 5 ) + 34
= ( x" + 5x )( x" + 5x + 10 ) + 34
= ( x" + 5x )" + 10( x" + 5x ) + 34
= x"( x" + 10x + 25 ) + 10x" + 50x + 34
= x"( x" + 10x ) + 25x" + 10x" + 50x + 34
= x^4 + 10x^3 + 35x" + 10x" + 50x + 34
再问：我要的是因式分解，大哥你这写的是啥？
再答：算一算，连续 4 个自然数的乘积 +10，得数是多少 x=0，1X2X3X4 +10 = 24 + 10 = 34 = 2X17 x=1，2X3X4X5 +10 = 120 +10 = 130 = 2X5X13 x=2，3X4X5X6 +10 = 360 +10 = 370 = 2X5X37 x=3，4X5X6X7 +10 = 840 +10 = 850 = 2X5X5X17 x=4，5X6X7X8 +10 = 1680+10 = 1690 = 2X5X13X13 x=5，6X7X8X9 +10 = 3024+10 = 3034 = 2X1517 让我们再根据实际数值，继续开动脑筋吧

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