Pamela, I'm afraid you might be misreading the arms in several ways.
Here is a link to your profile for Ada so others can follow along: Unknown Profile
It is clear that this depiction of the arms is not contemporary with Ada. These arms come from a later period, when heraldry was in decline (as defined by our modern notions) and there was great antiquarian interest in showing all the quarters a person was entitled to even if that person never used them.
If you are familiar with heraldry, you will see that this is the impaled shield of a married man. His arms are on the left with 18 "quarters" and his wife's arms are on the right with 9 "quarters".
The arms do not "scream" that this man was a son of Ada. Far from it.
In hundreds of years of heraldic writing, I don't think anyone has found a way to describe the system simply without using charts. But, in brief arms were inherited by a man's sons. If he had no sons, then his daughter (called a heraldic heiress) could pass the arms to her children but quartered with the arms of the children's father. The father's arms would be in the 1st and 4th quarters, and the mother's arms would be in the 2nd and 3rd quarters.
The next time someone in that male line married a heraldic heiress his children would put their mother's arms in the 3rd quarter. The next time someone in that male line married a herald heiress his children would put their first heiress ancestor's arms in the 2nd quarter, the next heiress ancestor in the 3rd quarter, and their mother's arms in the 4th quarter. The next time someone in that male line married a heraldic heiress his children would increase the number of "quarters" to 6, etc.
The system gets a little more complicated if any of the heiress ancestors had a coat of arms that was already quartered. In that case her shield was broken up, and all of her quarters appear in order after her paternal arms.
It's complicated to understand but once you get it, it very easy to read these quartered shields, particularly in conjunction with a pedigree and a basic reference book about the arms of different families.
The illustration you have shows Brereton in the 1st quarter. So these arms belong to a Brereton man. The next quarter appears to be Scotland. The next quarter appears is le Scot.
So, you know this man is descended from a female le Scot who had no brothers, but not how or how far back.
Other than the Georgian style of these arms (and the red rose of Lancaster), there are two other indications that this is a later antiquarian composition.
First, Scotland appears before le Scot. This is a dead give away. In the 13th century it would never occur to anyone that John le Scot had a right to the arms of Scotland in addition to the actual arms of his father. This type of academic elaboration came much later.
Second, even a moment of reflection should show you that Ada did not herself bring in all 17 extra quarters, so this is not the generation of her children. Instead, the position of le Scot in these quarterings shows that the le Scot heiress (whoever she was) was the most distant of several heiress ancestors.
The bottom line for your purposes is that this field is very late. It is evidence only about the belief in a le Scot descent in the generation when it was made. Since English genealogy was a mess up to the time of Horace Round, with many fanciful claims, I wouldn't credit these arms without corroborating evidence.
A fun project you might want to take on is to identify each of the quarters in the shield, then match the claim for inclusion to paper genealogy. If you use Papworth's Ordinary and work from a good genealogy it would be interesting and not very hard.