Final answer:
The given rational expression \[ \frac{x(xt-1)}{(x+3)(x-2)} \] is improper since both numerator and denominator have the same degree. To rewrite it as the sum of a polynomial and a proper rational expression, polynomial division is generally used. However, due to the inclusion of 't' in the numerator, the division is not straightforward without additional information.
Step-by-step explanation:
To determine if a rational expression is proper or improper, we compare the degrees of the numerator and the denominator. A rational expression is considered proper if the degree of the numerator is less than the degree of the denominator. It is improper if the degree of the numerator is greater than or equal to the degree of the denominator. The given rational expression is:
\[ \frac{x(xt-1)}{(x+3)(x-2)} \]
The degree of the numerator is 2, because it's in the form x(xt) which simplifies to x^2t. The degree of the denominator is also 2, as it's in the form (x)(x) which simplifies to x^2. Thus, with both numerator and denominator having the same degree, the given expression is an improper rational expression.
To rewrite it as the sum of a polynomial and a proper rational expression, we would have to perform polynomial long division or use synthetic division if possible. However, since the numerator includes a term with variable 't' as well, which complicates direct division with the denominator. In this case, we cannot perform the division unless the variable 't' is treated as a constant or given a specific numerical value.
Thus, without additional information about 't', we are unable to rewrite this improper fraction in the desired format.