The accompanying "granddaddy" chart was written when our textbook was 
using "x" as the independent variable.  I have not taken the trouble
to rewrite it just to replace "x" with "t", or to replace the degree
"n" of the indicated polynomials with the letter "m" in the current edition
of the textbook.

This chart just describes "MUC-eligible" functions that could
appear on the right-hand side of a linear, constant-coefficient DE, 
*for a fixed "alpha + i beta"*. The most general such function g is at the
left of the diagram.  Often we see simpler functions that may not look like
this most general g, but are special cases of it.  The diagram indicates
the relationships among the general g and its special cases.
The diagram does NOT show the MUC-form particular solution; the diagram
was already busy enough without that. 

Note also that, more generally, an "MUC-eligible" function g can be
any *linear combination* of functions with different "alpha + i beta"s.
(For functions with the *same* "alpha + i beta", always group them together
into the form shown at the left of the chart. Otherwise you will end up with
more undetermined coefficients than are needed, and you would get confused
in situations in which you needed to find the values of these coefficients.)