Using determinant properties

Row operations. $L_2 \leftarrow L_2 - L_1$, $L_3 \leftarrow L_3 - L_1$:
$$\Delta = \begin{vmatrix} 1 & a & a^2 \\ 0 & b-a & b^2-a^2 \\ 0 & c-a & c^2-a^2 \end{vmatrix}.$$

Factoring. $b^2-a^2 = (b-a)(b+a)$ and $c^2-a^2 = (c-a)(c+a)$, so
$$\Delta = \begin{vmatrix} 1 & a & a^2 \\ 0 & b-a & (b-a)(b+a) \\ 0 & c-a & (c-a)(c+a) \end{vmatrix}.$$
Factor $(b-a)$ from the second row and $(c-a)$ from the third:
$$\Delta = (b-a)(c-a)\begin{vmatrix} 1 & a & a^2 \\ 0 & 1 & b+a \\ 0 & 1 & c+a \end{vmatrix}.$$

Expansion. Expanding along the first column:
$$\Delta = (b-a)(c-a)\begin{vmatrix} 1 & b+a \\ 1 & c+a \end{vmatrix} = (b-a)(c-a)(c+a-b-a) = (b-a)(c-a)(c-b).$$

Conclusion. This is the Vandermonde determinant:
$$\Delta = (b-a)(c-a)(c-b).$$