RowReduce[{
{1, 3, -2, 2}, 
{0, 
          1, 1, -5}, 
{1, 2, -3, 
          7}, 
{-2, -8, 
          2, -1}
}]

MatrixForm[
      RowReduce[{
{1, 3, -2, 2}, 
{0, 
            1, 1, -5}, 
{1, 2, -3, 
            7}, 
{-2, -8, 
            2, -1}
}]]

Unfortunatelly Mathematica can not work with variables

MatrixForm[
      RowReduce[{
{1, 3, -2, 2, 
            b1}, 
{0, 1, 1, -5, 
            b2}, 
{1, 2, -3, 7, 
            b3}, 
{-2, -8, 2, -1, 
            b4}
}]]

The last result is not true. There are values of b1, b2, b3, b4 for which the system above is consistent. To find that out we have to do the row reduction by hand.  Below I will do it by hand but in Mathematica

MatrixForm[{
{1, 3, -2, 2, 
          b1}, 
{0, 1, 1, -5, 
          b2}, 
{1, 2, -3, 7, 
          b3}, 
{-2, -8, 2, -1, 
          b4}
}]

MatrixForm[{
{1, 3, -2, 2, 
          b1}, 
{0, 1, 1, -5, 
          b2}, 
{1, 2, -3, 7, b3} - {1, 3, -2, 2, 
            b1}, 
{-2, -8, 2, -1, b4} + 
          2 {1, 3, -2, 2, b1}
}]

{
{1, 3, -2, 2, b1}, 
{0, 1, 
        1, -5, 
        b2}, 
{1, 2, -3, 7, b3} - {1, 3, -2, 2, 
          b1}, 
{-2, -8, 2, -1, b4} + 
        2 {1, 3, -2, 2, b1}
}

MatrixForm[{
{1, 3, -2, 2, 
          b1}, 
{0, 1, 1, -5, 
          b2}, 
{0, -1, -1, 5, -b1 + b3} + {0, 
            1, 1, -5, b2}, 
{0, -2, -2, 3, 
            2 b1 + b4} + 
          2 {0, 1, 1, -5, b2}
}]

{
{1, 3, -2, 2, b1}, 
{0, 1, 
        1, -5, 
        b2}, 
{0, -1, -1, 5, -b1 + b3} + {0, 1, 
          1, -5, b2}, 
{0, -2, -2, 3, 
          2 b1 + b4} + 
        2 {0, 1, 1, -5, b2}
}

MatrixForm[{
{1, 3, -2, 2, 
          b1}, 
{0, 1, 1, -5, 
          b2}, 
{0, 0, 0, -7, 
          2 b1 + 2 b2 + b4}, 
{0, 0, 0, 
          0, -b1 + b2 + b3}
}]

The last matrix in row echelon form.  We see that the corresponding system is consistent if -b1+b2+b3 = 0. Let us verify that

MatrixForm[
      RowReduce[{
{1, 3, -2, 2, 
            5}, 
{0, 1, 1, -5, 
            2}, 
{1, 2, -3, 7, 
            3}, 
{-2, -8, 2, -1, 
            7}
}]]

True!