Prove that:
`{:|(a,b-c,c+b),(a+c,b,c-a),(a-b,b+a,)|=(a+b)+c)(a^2+b^2+c^2)`

Without expanding, prove the following |(a,b-c,c-b),(a-c,b,c-a),(a-b,b-a,c)|=(a+b-c)(b+c-a)(c+a-b)

Prove that: {:|(b+c,a-b,a),(c+a,b-c,b),(a+b,c-a,c)|=3abc -a^3-b^3-c^3

Prove that: |{:(-2a, a+b, a+c), (b+a,-2b,b+c),(c+a, c+b,-2c):}|=4(a+b)(b+c)(c+a)

Prove that: {:|(a^2,a,b+c),(b^2,b,c+a),(c^2,c,ab)| = -(a+b+c)(a-b)(b-c)(c-a)

Prove that |{:(b+c,a,a),(b,c+a,b),(c,c,a+b):}| = 4 abc

Prove that: |[a+b+2c,a,b],[c,b+c+2a,b],[c,a,c+a+2b]|= 2(a+b+c)^3

Prove that |((b+c)^2,ab,ca),(ab,(a+c)^2,bc),(ac,bc,(a+b)^2)|=2abc(a+b+c)^3

Without expanding, prove the following |(b+c,c+a,a+b),(c+a,a+b,b+c),(a+b,b+c,c+a)|=2(a+b+c)(ab+bc+ca-a^2-b^2-c^2)

Prove that |{:((b+c)^(2), a^(2), bc),((c+a)^(2), b^(2), ca),((a+b)^(2), c^(2), ab):}|= (a-b) (b-c)(c-a)(a + b+c) (a^(2) + b^(2) + c^(2)) .

Prove that |[b+c,a,a],[b,c+a,b],[c,c,a+b]| = 4abc