I've worked some problems, but I keep on getting stuck. Can you please help me?

Write the partial fraction decomposition of the rational expression.

1. (7x^2-x-12)/(x)(x+1)(x-1)

Here is what I did:

(x)(x+1)(x-1)[(7x^2-x-12)/(x)(x+1)(x-1)] = (x)(x+1)(x-1)[(a)/(x) + (b)/(x+1) + (c)/(x-1)]

Then I find the terms that can cancel out and I get:

7x^2-x-12= a(x+1)(x-1) + b(x-1)x+ c(x+1)x

7x^2-x-12=ax^2-a+bx-b+bx+cx+c+cx

7x^2-x-12= ax^2-a+bx^2-b+cx^2+c

Then I get the first equation: a+b=7, this is where I get stuck.

3. (9x^2-x-14)/(x^3-x)

(x)(x+1)(x-1)[(9x^2-x-14)/(x)(x+1)(x-1)= (x)(x+1)(x-1)[ (a)/(x) + (b)/(x+1)+ (c)/(x-1)

9x^2-x-14= a(x+1)(x-1)+ b(x-1)x + c(x+1)x

9x^2-x-14= ax^2-a-bx^2-b+cx^2+c

9x^2-x-14= ax^2-bx^2+cx^2-a-b+c

The equations:

a-b+c=9

b=0

b+c=-1

c=-14

Write the partial fraction decomposition of the rational expression.

11. (6x+5)/((x-9)^2)

(x-9)^2[(6x+5)/((x-9)^2)]= (x-9)^2[(a)/(x-9) + (b)/((x-9)^2))

(6x+5)/(x-9)(a+b)

(6x+5)/ax+bx-9a-9b

the equations:

a+b=6

-9a-9b=5

To get a variable to cancel, I mulitplied the first equation by 9 and got:

9a+9b=54

-9a-9b=5

both 9a and 9b cancel out and I just 59.

13. (5x^2-5x+7)/((x-1)^3)

((x-1)^3)(5x^2-5x+7)/((x-1)^3)= ((x-1)^3))[ (a)/(x-1) + (b)/((x-1)^2)+ (c)/((x-1)^3)

(5x^2-5x+7)=((x-1)^2)(a) + (x-1)(b) +c

(5x^2-5x+7)=ax^2+a+bx-b+c

The equations I couldnt get on this problem.

15. (x+4)/(x^3-2x^2+x)- This one I dont understand.

Write the partial fraction decomposition of the rational expression.

17. (12x+3)/ (x^3-1) - This one I dont understand

20. (12x^2-12x+6)/(x-3)(x^2+4)

6(2x^2-2x+1)=(x-3)(x^2+4) [(a)/(x-3) + (bx+c)/(x^2+4)

6(2x^2-2x+1)= a(x^2+4) + bx+c(x-3)

6(2x^2-2x+1) = ax^2+4a+bx^2-3bx+cx-3c

The equations:

a+b=6

4a-3b (this is where I get stuck)

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants.

26. (7x-3)/((x^2+x-7)^2)

I got: (ax+b)/(x^2+x-7) + (cx+d)/((x^2+x+7)^2)

29. (2x+1)/(x-6)((x^2+x-4)^2)

I got: (a)/(x-6) + (bx+c)/(x^2+x-4)

Write the partial fraction decomposition of the rational expression.

31. (x^2+4x-1)/((x^2+3)^2)

x^2+4x-1 = ax +b(x^2+3)+cx+b/((x^2+3)^2)

x^2+4x-1= ax+b(x^2+3)+cx+d

x^2+4x-1=ax^3+3a+bx^2+3b+cx+d

x^2+4x-1=ax^3+bx^2+ (3a+c)x + 3b+d

The equations:

a=1

b=0

3a+c=4

3b+d=-1

When I come up with the fraction I get: (x+4)/(x^2+3) + (-4x - 12)/((x^2+3)^2))

33. (4x^3+ 4x^2)/((x^2+5)^2))

I did: (4x^3+ 4x^2)= x^2+5 (ax+b) + cx+d

(4x^3+ 4x^2)= ax^3+bx^2+ 5a+5b+cx+d

The equations:

a=4

b=4

5a+c=0

5b+d=0

The final answer I got: (4x+4)/(x^2+1) + (-20x-20)/((x^2+5)^2)