Let n be the number of small squares in each way, originally.


After adding one row each way, there are (n+1) small squares in each way. 


The area of the original patchwork quilt is  n^2;
The area of the patchwork quilt after enlarging is  (n+1)^2;
the difference of the two areas is 177. So, we can write this equation 

    (n+1)^2 - n^2 = 177.


Simplify this equation and find n

    n^2 + 2n + 1 - n^2 = 177

          2n + 1 = 177

          2n = 177 - 1 = 176

           n           = 176/2 = 88.


ANSWER.  The number of small squares originally is 88^2 = 7744.

A square patchwork quilt is made of many small squares.  The quilt is too small to place on designated furniture.  Quilt is to be enlarged square one row each way.  177 more squares are made and are just enough to increase to desired size.  Determine number of small squares in original.

Not sure how to solve.

Let number of small squares in each row, be S
Then, S will also be the number of small squares in each column
Area of ORIGINAL quilt would then be S x S = S2

With 1 small square being added to each row, number of small squares in each row would increase to
S + 1. Likewise, number of small squares in each COLUMN would also increase to S + 1, thereby making
the quilt's new area, (S + 1)2.

The difference in the original area and the new area is a total 177 small squares. 
Therefore, we get the following: (S + 1)2 - S2 = 177
                              S2 + 2S + 1 - S2 = 177
                                        2S + 1 = 177
                                            2S = 176
Original number of small squares in each row/column, or 
Original number of small squares on quilt = 88 x 88, or 882 = 7,744