Hermite interpolating polynomial?

Hello, I have just been given a question that I don't understand and after looking over my course notes I can't find anything resembling it. So could anyone could show me how to solve it, explain what exactly this question is about, or point me in the direction of a site that explains it clearly without too much reliance on other mathematics topics? I mainly just want to know the technique so I can do this thing. **Any help whatsoever** would be __greatly__ appreciated!

Let

Let , be two distinct points at which the function and its first derivative are defined and assume that the second derivative is also defined at the point . A Hermite interpolating quartic polynomial of the form

can be constructed for the function from the given five data values by determining the unknown coefficients using the conditions:

and

Construct a Hermite quartic polynomial that interpolates the function at the eigenvalues of by solving an appropriately defined linear system whose solution provides the coefficients , . Use this polynomial to construct a matrix function approximation for .