I need help with partial fractions!

i feel as if i know the steps somewhat but i can't d=seems to find out the variables for A and B probably because my basics aren't so good. i am hoping to nail this by the end of the week so any suggestions to how i can improve the part that confuses me is the system finding here are some examples that i don't get i would really appreciate it if any of you guys can solve them step by step but also give me some other resources i can look at for further comprehension. Thanks ♥
here are the problems i'll put parentheses to make it look more clear
(x-3)/(x^3+x^2)
(x^2+x+2)/(x^2+2)^2
(5-x)/(2x^2+x-1)
(x^3)/(x+2)^2(x-2)^2
when writing the [partial fraction decomposition of expression (x^3+x-2)/(x^2-5x-14) the step is to factor the denominator.
(x)/(16x^4-1)
(x^2-4x)/(x^2+x+6)
the improper ones i think are a lot harder but fairly the same except for 2 additional steps which includes the written long division
sorry for all the questions but 10 points are rewarded to the best one "figures"

(x²+x+2) / (x²+2)² = (Ax+B)/(x²+2) + (Cx+D) /(x²+2)²

multiply by (x²+2)²

(x²+x+2) = (Ax+B)(x²+2) + (Cx+D), distribute group them according to degree of x

(x²+x+2) =Ax^3+ Bx²+ x(2A+C) +2B+D

equate the coefficients

1=B
0=A
1=(2A+C)
2=2B+D

solve the system
1=B
0=A
C=1,
D=0
plug back into the partial fraction
(x²+x+2) / (x²+2)² = (0x+1)/(x²+2) + (1x+0) /(x²+2)²

x²+x+2) / (x²+2)² = 1/(x²+2) + x/(x²+2)²

"when writing the [partial fraction decomposition of expression (x^3+x-2)/(x^2-5x-14) the step is to factor the denominator.
(x)/(16x^4-1)
(x^2-4x)/(x^2+x+6)" ??????? "(x)/(16x^4-1)": what is this nonsense? "(x^2-4x)/(x^2+x+6)": or this?


The factorization of x^2-5x-14 is (x+2)(x-7)

The partial fraction expansion, depending on what you want to get, is either

(1/9){[2x^2/(x+2)] +[(7x^2+9)/(x-7)]}
or
x-5 +(1/9){[2/(x-7)] - [92/(x+2)]}

Here, you have,

(x-3)/(x^3+x^2) = (x-3) / x^2(x+1) = A/x +B/x^2 +D/(x+1) .......... (say)

Multiply both-sides by x^2(x+1), we get,
x-3 = Ax(x+1) +B(x+1) +Dx^2,
SO,
A+D = 0, A = 1, and B = -3, SO, D = -1
Hence,
(x-3)/(x^3+x^2) = 1/x -3/x^2 -1/(x+1) >====================< ANSWER

Similarly Solve the Others............


SOLving, C=-3, A+B = 4, D = -4