Đáp án ` + ` Giải thích các bước giải:

`#\text{Day 1}`

` a) `

` A = a(b^2 + c^2 - a^2) + b(c^2 + a^2 - b^2) `

` = ab^2 + ac^2 - a^3 + bc^2 + ba^2 - b^3 `

` = (ab^2 + ba^2) + (ac^2 + bc^2) - (a^3 + b^3) `

` = ab(a + b) + c^2(a + b) - (a + b)(a^2 - ab + b^2) `

` = (a + b)[ab + c^2- (a^2 - ab + b^2)] `

` = (a + b)(2ab + c^2 - a^2 - b^2) `

` = (a + b)[c^2 - (a^2 - 2ab + b^2)] `

` = (a + b)(c - a + b)(c + a - b) ` 

$\\$

` b) `

` B = a(b^2 + c^2) + b(a^2 + c^2) + c(a^2 + b^2) + 2abc `

` = ab^2 + ac^2 + ba^2 + bc^2 + ca^2 + cb^2 + 2abc `

` = (ba^2 + ab^2 + abc) + (ca^2 + ac^2 + abc) + (bc^2 + cb^2) `

` = ab(a + b + c) + ac(a + c + b) + bc(b + c) `

` = a(a + b + c)(b + c) + bc(b + c) `

` = (b + c)[a(a + b + c) + bc] `

` = (b + c)(a^2 + ab + ac + bc) `

` = (b + c)[(a^2 + ab) + (ac + bc)] `

` = (b + c)(a + b)(a + c) `

` = (a + b)(b + c)(c + a) `

$\\$

` c) `

` C = (x + y)^5 - x^5 - y^5 `

` = (x + y)^5 - (x^5 + y^5) `

` = (x + y)^5 - (x + y)(x^4 - x^3y + x^2y^2 - xy^3 + y^4) `

` = (x + y)[(x + y)^4 - (x^4 - x^3y + x^2y^2 - xy^3 + y^4)] `

` = (x + y)[(x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4) - (x^4 - x^3y + x^2y^2 - xy^3 + y^4)] `

` = (x + y)(x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4 - x^4 + x^3y - x^2y^2 + xy^3 - y^4) `

` = (x + y)(5x^3y + 5x^2y^2 + 5xy^3) `

` = (x + y) * 5xy(x^2 + xy + y^2) `

` = 5xy(x + y)(x^2 + xy + y^2) `