$b)$ $B = (a + b + c)^3 - (a + b - c)^3 - (b + c - a)^3 - (c + a - b)^3$

* Đặt $x = a + b - c$, $y = b + c - a$, $z = c + a - b$. Ta có:

$x + y + z = (a + b - c) + (b + c - a) + (c + a - b) = a + b + c$

* Khi đó, $B$ trở thành:

$(x + y + z)^3 - x^3 - y^3 - z^3$

$= [(x + y) + z]^3 - x^3 - y^3 - z^3$

$= (x + y)^3 + 3(x + y)^2z + 3(x + y)z^2 + z^3 - x^3 - y^3 - z^3$

$= x^3 + 3x^2y + 3xy^2 + y^3 + 3(x + y)^2z + 3(x + y)z^2 + z^3 - x^3 - y^3 - z^3$

$= 3xy(x + y) + 3(x + y)z(x + y + z)$

$= 3(x + y)[xy + z(x + y + z)]$

$= 3(x + y)(xy + xz + yz + z^2)$

$= 3(x + y)[x(y + z) + z(y + z)]$

$= 3(x + y)(y + z)(z + x)$ $(1)$

* Trả ẩn:

$x + y = (a + b - c) + (b + c - a) = 2b$

$y + z = (b + c - a) + (c + a - b) = 2c$

$z + x = (c + a - b) + (a + b - c) = 2a$

* Thay $x + y = 2b$; $y + z = 2c$; $z + x = 2a$ vào $(1)$ ta được:

$= 3 \cdot 2b \cdot 2c \cdot 2a$

$= 24abc$

$c)$ $C = ab(a + b) + bc(b + c) + ca(c + a) - a^3 - b^3 - c^3 - 2abc$

* Đặt $x = a + b$, $y = b + c$, $z = c + a$. Ta có:

$C = abx + bcy + caz - a^3 - b^3 - c^3 - 2abc$

$C = a^2b + ab^2 + b^2c + bc^2 + c^2a + ca^2 - a^3 - b^3 - c^3 - 2abc$

$C = (a + b)(b + c)(c + a) - (a^3 + b^3 + c^3 + 3abc)$

$C = xyz - (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) - 6abc$

$C = xyz - \frac{1}{2}(x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) - 6abc$

$C = (a + b - c)(b + c - a)(c + a - b)$

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