\(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)
\(=\left(x+2\right)\left(x+5\right)\left(x+3\right)\left(x+4\right)-24\)
\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)
\(=\left(x^2+7x+10\right)^2+2\left(x^2+7x+10\right)-24\)
\(=\left(x^2+7x+11\right)^2-25\)
\(=\left(x^2+7x+6\right)\left(x^2+7x+16\right)\)
\(=\left(x+1\right)\left(x+6\right)\left(x^2+7x+16\right)\)
\((x+2)(x+3)(x+4)(x+5)-24\\=[(x+2)(x+5)]\cdot[(x+3)(x+4)]-24\\=(x^2+7x+10)(x^2+7x+12)-24\)
Đặt \(y=x^2+7x+10\), khi đó biểu thức trở thành:
\(y(y+2)-24\\=y^2+2y-24\\=y^2+2y+1-25\\=(y+1)^2-5^2\\=(y+1-5)(y+1+5)\\=(x^2+7x+10+1-5)(x^2+7x+10+1+5)\\=(x^2+7x+6)(x^2+7x+16)\\=(x^2+x+6x+6)(x^2+7x+16)\\=[x(x+1)+6(x+1)](x^2+7x+16)\\=(x+1)(x+6)(x^2+7x+16)\\Toru\)
(x + 2)(x + 3)(x + 4)(x + 5) - 24
= [(x + 2)(x + 5)][(x + 3)(x + 4)] - 24
= (x² + 5x + 2x + 10)(x² + 4x + 3x + 12) - 24
= (x² + 7x + 10)(x² + 7x + 12) - 24 (1)
Đặt t = x² + 7x + 10
(1) = t.(t + 2) - 24
= t² + 2t - 24
= t² - 4t + 6t - 24
= (t² - 4t) + (6t - 24)
= t(t - 4) + 6(t - 4)
= (t - 4)(t + 6)
= (x² + 7x + 10 - 4)(x² + 7x + 10 + 6)
= (x² + 7x + 6)(x² + 7x + 16)
= (x² + x + 6x + 6)(x² + 7x + 16)
= [(x² + x) + (6x + 6)](x² + 7x + 16)
= [x(x + 1) + 6(x + 1)](x² + 7x + 16)
= (x + 1)(x + 6)(x² + 7x + 16)
