Two matrix determinant questions

I have two problems I've been assigned to assist in studying for an exam, but I can't figure out how to proceed. The first one is:

Prove the following:

|1 a bc| --|1 a a^2|----------------- |1 a a^2|

|1 b ac| = |1 b b^2| = (c-a)(b-a)(c-b)|0 1 b+a |=(c-a)(b-a)(c-b)

|1 c ab| --|1 c c^2|----------------- |0 0 1 |

Using only these rules:

1: If each element of a one row or column of a determinant is multiplied by a number k, the value of the determinant is multiplied by k.

2: The value of a determinant is zero if

a: all elements of one row or colum are zero, or if

b: two rows or columns are identical, or if

c: two rows or colums are proportional

3: If two rows or colums of a determinant are interchanged, the value of the determinant changes sign.

4: The value of a determinant is unchanged if

a: rows are written as columns and columns as rows, or if

b: we add to each element of one row or column, k times the corresponding element of another row or column, where k is any number.

My second question is more general.

When trying to determine the determinant of a matrix containing polynomials by hand, is there any way to do it other than brute forcing it with tons of polynomial multiplication?