`|[x^2+a^2,ab,ac] , [ab,x^2+b^2,bc] , [ac,bc,x^2+c^2]|=`

Using the properties of determinant, show that : |[a^2+1,ab,ac],[ab,b^2+1,bc],[ac,bc,c^2+1]| = 1+a^2+b^2+c^2

Prove that: |[a^2+1,ab,ac],[ab,b^2+1,bc],[ac,bc,c^2+1]|=|[a^2+1,b^2,c^2],[a^2,b^2+1,c^2],[a^2,b^2,c^2+1]|=1+a^2+b^2+c^2

Prove that: |[-a^2, ab,ac],[ba,-b^2,bc],[ca,cb,-c^2]|=4a^2b^2c^2

Prove that: |[a^2,bc,ac+c^2],[a^2+ab,b^2,ac],[ab,b^2+bc,c^2]|=4a^2b^2c^2

If a,b,c are real numbers, then find the intervals in which f(x) = {:|(x+a^2,ab,ac),(ab,x+b^2,bc),(ac,bc,x+C^2)| is striclty increasing or decreasing.

If f(x)={:abs((x+a^(2),ab,ac),(ab,x+b^(2),bc),(ac,bc,x+c^(2))):} , then find f'(x).

Prove that |{:(b^(2)+c^(2),ab,ac),(ab,c^(2)+a^(2),bc),(ac,bc,a^(2)+b^(2)):}| = 4a^(2)b^(2)c^(2)

Prove that |(bc-a^2, ca-b^2, ab-c^2),(ca-b^2, ab-c^2, bc-a^2),(ab-c^2, bc-a^2, ca-b^2)| is divisible by a+b+c. Also find the value of the quotient.