<s>[INST] Proving determinants using properties of determinants $$\begin{vmatrix}
1 & a^2+bc  & a^3\\
1 & b^2+ca  & b^3\\
1 & c^2+ab  & c^3 
\end{vmatrix}
 =  (a-b)(b-c)(c-a)(a^2+b^2+c^2)$$
we have to solve this by using the properties of determinants without actually expanding the determinant. I am Unable to think which calculation to apply so was hoping for an hint.
Edit: just tried the problem and here is how I have done it
$$\begin{vmatrix}
1 & a^2+bc  & a^3\\
1 & -(a^2+b^2)+c(a-b)  & -(a^3-b^3)\\
1 & c^2-a^2-b(c-a)  & c^3-a^3 
\end{vmatrix}$$
then
$$\begin{vmatrix}
1 & a^2+bc  & a^3\\
0 & -((a-b)(a+b))+c(a-b)  & -((a-b)(a^2+ab+b^2))\\
0 & (c-a)(c+a)-b(c-a)  & (c-a)(c^2+ca+a^2) 
\end{vmatrix}$$
then
$$(a-b)(c-a)\begin{vmatrix}
1 & a^2+bc  & a^3\\
0 & a+b+c  & -(a^2+ab+b^2)\\
0 & a-b+c  & (c^2+ca+a^2) 
\end{vmatrix}$$
then
$$(a-b)(c-a)\begin{vmatrix}
  a+b+c  & -(a^2+ab+b^2)\\
  a-b+c  & (c^2+ca+a^2) 
\end{vmatrix}$$
then 
$$(a-b)(c-a) *  (  (a+b+c)(c^2+ca+a^2)-(a-b+c)(-(a^2+ab+b^2))  )$$
Here I am Confused on how to multiply them and get the answer
note: typed because I own a very bad handwriting 
Thank you every one for your help
 [/INST] The solution-attempt in the question is excellent (after your edit)! 
But unfortunately you made a small mistake. After taking out the factor $(a-b)(c-a)$, the middle element should be $c-a-b$, not $a+b+c$. If you fix that, the remaining determinant can be expanded and easily factorised.
</s>