Discriminant of the polynomial $f(x)=4x^3-ax-b$ 
Definition. The discriminant of the polynomial $f(x)=4(x-x_1)(x-x_2)(x-x_3)$ is the product $16\{(x_2-x_1)(x_3-x_2)(x_3-x_1)\}^2$. 

How to prove that the discriminant of $f(x)=4x^3-ax-b$ is $a^3-27b^2$.
Any help would be appreciated.

You should read about the sylvester Sylvester matrix.
To be precise, $f(x) = 4x^3-ax−b$ gives $f'(x) = 12 x^2 - a$. So the discriminant is the determinant
$$ \Delta(a,b) = \left|\begin{array}{ccccc} 
4 & 0 & -a & -b & 0 \\
0 & 4 & 0 & -a & -b\\
12 & 0 & -a & 0 & 0\\
0 & 12 & 0 & -a & 0\\
0 & 0 &12 & 0 & -a
\end{array}\right|
 = -64(a^3 - 27b^2).$$
