Gensler's star test is a simplified method for determining the validity of a syllogism proposed in 1973. According to the test, one stars (asterisks) the first (capital) letter after "All", and all letters after "not" or "No". The syllogism is valid if and only if every capital letter is starred exactly once and there is exactly one star on the right-hand side.
The test has limitations, however. It only applies to categorical syllogisms, and even to those only if we make a modern assumption that "All" does not carry existential import. In other words, "All A are B" does not imply that there are any A-s. But the OP example uses the Darapti syllogism form: All A are B, all A are C, therefore, some B are C, which is only valid with existential import, see Why is the darapti syllogism invalid? Indeed, if there are no teachers we can not conclude that some intelligent people are well paid based on the OP premises. So the Darapti form is invalid, just as the star test predicts.
In modern logic we translate "All A are B" into
∀x(A(x) → B(x)). If there are no A-s then A(x) is always false, hence the implication is always true, and so is the quantified statement. In other words, modern logic does not give existential import to "All", and Gensler's test adopts this modern interpretation. It was different in classical times, Aristotle himself considered Darapti a valid form.
If we go with Aristotle, Darapti is not the only example where the star test fails. The same applies to some other forms of the third and fourth figure. Yildirim gives a detailed analysis of the star test and limits of its validity in
Gensler's Star Test and Some Examples of its Application:
"Gensler’s star test is a useful and functional method for checking the validity of syllogisms. But we should note that this method is useless in the conditional or the hypothetical syllogism, compound syllogisms such as the compound conditional or the compound syllogism «involving a contradiction» (qıyas al-khalf), and Darapti or Felapton types of syllogisms which are third figure (Middle terms of premises are subjects of the premises). And also we can add Fesapo or Bramantip types of syllogisms of fourth figure, enthymeme, etc."
An alternative to Gensler's test is the use of Venn diagrams. Grennan describes how to modify their use when the existential import for "All" is assumed in Informal Logic, p.119.