<s>[INST] How to calculate the determinant of a $4 \times 4$ matrix with multiple variables? What is the determinant: $$ \begin{vmatrix}1& a & a^2 & a^4 \\ 1 & b & b^2 & b^4 \\ 1 & c & c^2 & c^4 \\1 & d & d^2 &d^4 \end{vmatrix} $$
Someone gave me the following hint 

Replace $d$ by a variable $x$; make use of the fact that the sum of the roots of a fourth-degree polynomial is equal to the coefficient of $x^3$

but I didn't get that.
 [/INST] As you already noted the determinant is a polynomial of 4th degree in $d$. 
This polynomial is zero if you replace $d$ by $a$, $b$, or $c$ (we have two identical rows). 
Moreover the coefficient of $d^3$ is zero which implies that the sum of the roots is zero. Hence the fourth root is $-(a+b+c)$.
Finally, by Vandermonde's determinant,  the coefficient of $d^4$ is
$$\det\begin{pmatrix}1& a& a^2\\1& b& b^2\\1& c& c^2
\end{pmatrix}=(a-b)(b-c)(c-a).$$
Therefore the required determinant is
$$(a-b)(b-c)(c-a)\cdot (d-a)(d-b)(d-c)(d+a+b+c).$$
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