Is this correct: Representation of a 4 to 9-node-quadrilateral element?
and this: which is the case of a multi-degree of freedom problem?
If you're talking about the hyphenation, I believe the first sentence should look like this:
Representation of a 4-to-9-node quadrilateral element.
Here, 4-to-9-node modifies quadrilateral, which in turn modifies element.
Which is the case of a multi-degree-of-freedom problem.
Because of the preposition of, degree-of-freedom modifies problem. However, multi also becomes part of that phrase.
With this one, if single were in front rather than multi it would be ambiguous unless the meaning were known in advance.
It could be a single instance of a degree-of-freedom problem:
A single degree-of-freedom problem.
Or, it could be a degree-of-freedom problem that has only a single degree:
A single-degree-of-freedom problem.
In order to understand which words are hyphenated, you need to understand the meaning of what's being expressed.
That not all elements that come before a final noun are hyphenated is explained in The Chicago Manual of Style (17th ed.), 7.87:
Multiple hyphens are usually appropriate for such phrases as an over-the-counter drug or a winner-take-all contest. If, however, the compound modifier consists of an adjective that itself modifies a compound, additional hyphens may not be necessary. The expressions late nineteenth-century literature and early twentieth-century growth are clear without a second hyphen.
There are a lot of other guidelines when it comes to specific phrases. Although the general guidelines are clear, there are a a lot of exceptions. Chicago actually has a very long table of word and phrases along with hyphenations for them.