Find determinant Find determinant of A.
$${det}A =\left( \begin{array}{cc}
1 & 1 & 1 & 1 & 1\\
a & b & c & d & x\\
a^2 & b^2 & c^2 & d^2 & x^2\\
a^3 & b^3 & c^3 & d^3 & x^3\\
a^4 & b^4 & c^4 & d^4 & x^4\end{array} \right)$$
I was thinking that I should reduce ones on top, but then I would be stuck with this:
$${det}A =\left( \begin{array}{cc}
b-a & c-a & d-a & x-a\\
b^2-a^2 & c^2-a^2 & d^2-a^2 & x^2-a^2\\
b^3-a^3 & c^3-a^3 & d^3-a^3 & x^3-a^3\\
b^4-a^4 & c^4-a^4 & d^4-a^4 & x^4-a^4\end{array} \right)$$
Any suggestions?

$\det A(x)$ is a polynomial in variable $x$ of degree $4$.
Notice that for $x \in \{a, b, c, d\}$ you obtain $\det A(x) = 0$ so $a, b, c, d$ are the four roots of the polynomial $\det A(x)$.
Therefore, $$\det A(x) = \alpha (x - a)(x - b)(x - c)(x-d)$$
for some $\alpha \in \mathbb{R}$.
In particular, we have $$\alpha \cdot abcd = \det A(0) = \begin{vmatrix}
1&1&1&1&1\\
a&b&c&d&0 \\
a^2&b^2&c^2&d^2&0 \\
a^3&b^3&c^3&d^3&0 \\
a^4&b^4&c^4&d^4&0
\end{vmatrix} = \begin{vmatrix}
a&b&c&d \\
a^2&b^2&c^2&d^2 \\
a^3&b^3&c^3&d^3 \\
a^4&b^4&c^4&d^4
\end{vmatrix} = abcd\begin{vmatrix}
1&1&1&1\\
a&b&c&d \\
a^2&b^2&c^2&d^2 \\
a^3&b^3&c^3&d^3 \\
\end{vmatrix}$$
Now again recursively, if we look at this determinant as a polynomial in the variable $d$ and then $c$ and then $b$, we obtain
$$\alpha \cdot abcd = \det A(0) = abcd \cdot (b-a)(c-a)(c-b)(d-a)(d-b)(d-c)$$
Therefore $\alpha = (b-a)(c-a)(c-b)(d-a)(d-b)(d-c)$ so:
$$\det A(x) = (b-a)(c-a)(c-b)(d-a)(d-b)(d-c)(x - a)(x - b)(x - c)(x-d)$$
