Aaron's home, the grocery store, and the library are all on the same road. the total distance from aaron's home to the library can be represented by the expression 54x2+310x+8 . the grocery store is located between his home and the library. the distance from the store to the library can be represented by the expression 13x2−47x+23 .

The distance from his home to the store is represented by

41x² + 357x - 15.

We know the distance from the store to the library, and the distance from his home to the library.  We also know the store is between the two.  To find the missing distance, we subtract the two polynomials we have:

(54x² + 310x + 8) - (13x² - 47x + 23)

Combining like terms:
54x² - 13x² = 41x²;
310x -- 47x = 310x + 47x = 357x;
8-23 = -15

This gives us 

41x²+357x-15