Decompose 3x-1/x^2-x into partial fractions.

Decompose 3x-1/x^2-x into partial fractions.

Decompose x-1/ 3x^2-14x+15 into partial fractions.

Decompose 8x+22/x^2+4x-5 into partial fractions

3x-1/x^2-x

solve (3x-1) / x(x-1) = a/x + b / (x-1)
multiply each term by x(x-1)
3x -1 = a(x-1) + bx
= ax +bx -a
= (a+b) x -a
so a= 1
a+b = 2
3+b = 2
b = -1
so (3x-1) /(x^2 -x) = 1/x - 1/ (x-1)

Your other questions are left to you to practice.