Đáp án:
a, Ta có :
`A = (x + 1)(x + 2)(x + 3)(x + 4) - 24`
` = [(x + 1)(x + 4)].[(x + 2)(x + 3)] - 24`
`= (x^2 + x + 4x + 4)(x^2 + 2x + 3x + 6) - 24`
`= (x^2 + 5x + 4)(x^2 + 5x + 6) - 24`
Dặt `t = x^2 + 5x + 5`
`=> A = (t - 1)(t + 1) - 24`
`= t^2 - 1 - 24`
`= t^2 - 25`
`= t^2 - 5^2`
`= (t - 5)(t + 5)`
`=> A = (x^2 + 5x + 5 - 5)(x^2 + 5x + 5 + 5)`
`= (x^2 + 5x)(x^2 + 5x + 10)`
b, Ta có :
`(x^2 + 2x)^2 + 9x^2 + 18x + 20`
`= (x^2 + 2x)^2 + 9(x^2 + 2x) + 20`
`= (x^2 + 2x)^2 + 2.(x^2 + 2x) . 9/2 + 81/4 - 1/4`
`= (x^2 + 2x + 9/2)^2 - (1/2)^2`
`= (x^2 + 2x + 9/2 - 1/2)(x^2 + 2x + 9/2 + 1/2)`
`= (x^2 + 2x + 4)(x^2 + 2x + 5)`
c, Ta có :
`B = (a + b + c)^3 - (a + b - c)^3 - (b + c - a)^3 - (c + a - b)^3`
`= [(a + b + c)^3 - (a + b - c)^3] - [(b + c - a)^3 + (c + a - b)^3]`
` = (a + b + c - a - b + c)^3 + 3(a + b + c)(a + b - c)[(a + b + c) - (a + b - c)] - [(b + c - a + c + a - b)^3 - 3(b + c - a)(c + a - b)(b + c - a + c + a - b)`
`= (2c)^3 + 3(a + b + c)(a + b - c)(a + b + c - a - b + c) - (2c)^3 + 3(b + c - a)(c + a - b)(b + c - a + c + a - b)`
`= 3(a + b + c)(a + b - c).2c + 3(b + c - a)(c + a - b).2c`
`= 6c.[(a + b + c)(a + b - c) + (b + c - a)(a + c - b)]`
Đặt `b + c - a = x` ; `c + a - b = y` ; `a + b - c = z`
`=> x + y + z = a + b + c`
`=> B = 6c.[(x + y + z).z + x.y]`
` = 6c(zx + zy + z^2 + xy)`
`= 6c[(zx + xy) + (zy + z^2)]`
`= 6c[x(z + y) + z(y + z)]`
`= 6c(x + z)(y + z)`
`= 6c(b + c - a + a + b - c)(c + a - b + a + b - c)`
`= 6c.2b.2a`
`= 24abc`
Giải thích các bước giải:
