For part (a), this is just development (Laplace expansion) of the determinant by the first row. Actually the $\det()$ factors should have alternating signs. Since the only occurrences of $x$ are in that first row, all the $\det()$ expressions are constants, and one gets a polynomial of degree at most $n-1$ (from the final term) in $x$.
For part (b), that $P(a_i)=0$ for $i=2,3,\ldots,n$ is just the fact that $P(a_i)$ equals the determinant of the matrix obtained by substituting $a_i$ for $x$, so from the original matrix $a_1$ has been replaced by $a_i$, and as this matrix has its rows $1$ and $i$ identical, its determinant vanishes. all this uses is that the Laplace expansion used commutes with such substitution. Furthermore a polynomial of degree at most $n-1$ with $n-1$ specified roots $a_2,\ldots,a_n$ can only be a scalar multiple of $(x-a_2)\ldots(x-a_n)$.
For part (c), this is just remarking that the $\det()$ in question is $(-1)^{n-1}$ times the determinant of the lower-left $(n-1)\times(n-1)$ submatrix, which determinant precisely matches the definition of $V_{n-1}(a_2,\ldots,a_n)$.
For part (d) write $(-1)^{n-1}\prod_{i=2}^n(x-a_i)=\prod_{i=2}^n(a_i-x)$ to get
$$
V(x,a_2,\ldots,a_n)=
(-1)^{n-1}V_{n-1}(a_2,\ldots,a_n)\prod_{i=2}^n(x-a_i)
=V_{n-1}(a_2,\ldots,a_n)\prod_{i=2}^n(a_i-x),
$$
and then set $x=a_1$ to get
$$
V(a_1,a_2,\ldots,a_n)
=V_{n-1}(a_2,\ldots,a_n)\prod_{i=2}^n(a_i-a_1),
$$
Part (e) applies induction on $n$ to $V_{n-1}(a_2,\ldots,a_n)$ (the starting case is $V_0()=1=\prod_{1\leq i<j\leq 0}1$, an empty product, or if you fear $n=0$ it is $V_1(a)=1=\prod_{1\leq i<j\leq 1}1$, still an empty product), to get
$$
V(a_1,a_2,\ldots,a_n)
=\left(\prod_{2\leq i<j\leq n}(a_j-a_i)\right)\prod_{j=2}^n(a_j-a_1)
=\prod_{1\leq i<j\leq n}(a_j-a_i).
$$
Best Answer
The Vandermonde determinant is equal to
$$\det \begin{pmatrix} 1 & 1 & 1 & ... & 1 \\ a_1 & a_2 & a_3 & ... & a_n \\ a_1^2 & a_2^2 & a_3^2 & ... & a_n^2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ a_1^{n-1} & a_2^{n-1} & a_3^{n-1} & ... & a_n^{n-1}\end{pmatrix} = \prod_{i<j}(a_i-a_j)$$
Therefore, by Cramer's rule the solution of your linear system is
$$x_k = \frac{\prod_{\substack{i<j \\ k \leftrightarrow b}}(a_i-a_j)}{\prod_{i<j}(a_i-a_j)}$$
in which I introduced some notation to indicate that in the upper product, the $a_k$'s have to be switched for the $b$'s.
As suggested by Martin, this simplifies further to
$$x_k = \prod_{i\neq k}\frac{a_i-b}{a_i-a_k}$$