You can't carry courses over from one school year to the next with HomeSchool Easy Records. The reason for this is it keeps the report card "straight" and tends to avoid confusion. A student is typically at a single grade level in a single school year and the report card covers a single year and grade level. But you can "work around" this by following this example. Keep in mind that you can't have the same course name at the same student level in two different school years. The issue of course carry-over applies mainly to the report card and the transcript.
Say the course is "Science 7" and you want to carry this course from year "1" to year "2". Let's also assume the student is in grade 7 in year 1 and grade 8 in year 2. In year 1, give this course a type of "O" (other) in the Course Setup screen. This will prevent this course at grade level 7 from showing up in the Transcript. Give a final grade and completion date to the course in year 1 -- you can use a final grade of "Inc" (incomplete). Now set the same course up for year 2 using a student level of 8, a type of "C" (credit), and some non-zero value for Credits. The course will show up only once on the transcript as being taken in grade 8 (but the course name will suggest it is a level 7 course). The report card for year 2 won't show the quarters from year 1 for this course. If you go back and print the report card for year 1, this course now won't show up at all since it is marked as type "other" (you'd have to temporarily call it a credit course again just for printing the old report card). When the course is completed in year 2, give it a final grade and completion date, and make sure it has Credits of greater than zero (or else it won't work into the GPA in the transcript). The final grade and credits will show up on the transcript and figure into the GPA.