`(x+2)(x+3)(x+4)(x+5)-24`
`=[(x+2)(x+5)][(x+3)(x+4)]-24`
`=(x^2+5x+2x+10)(x^2+4x+3x+12)-24`
`=(x^2+7x+10)(x^2+7x+12)-24`
`=(x^2+7x+11-1)(x^2+7x+11+1)-24`
`=(x^2+7x+11)^2-1-24`
`=(x^2+7x+11)^2-25`
`=(x^2+7x+11)^2-5^2`
`=(x^2+7x+11-5)(x^2+7x+11+5)`
`=(x^2+7x+6)(x^2+7x+16)`
`=(x^2+x+6x+6)(x^2+7x+16)`
`=(x+1)(x+6)(x^2+7x+16)`