Aardvark

Here is the easy explanation:

Notice first that the polynomial fraction on the left has a degree of -1. How? The numerator has a degree 2 and the denominator 3. So the degree is 2 - 3 = -1.

So each polynomial fraction on the right must have a degree of -1.

Look at the first polynomial fraction A/(x - 2). The numerator is a zero degree polynomial (aka, a constant) and denominator is a degree 1 polynomial (x - 2).

Now, look at the denominator of the 2nd fraction (x^2 + 2x + 4). This is a 2nd degree polynomial. If the whole fraction must have a degree of -1, then the numerator must be a 1st degree polynomial (1 - 2 = -1). So the 1st degree numerator is Bx+C.

Alternately, imagine what happens when the fractions on the right are added. You should get a numerator with degree 2, since the fraction on the left has a numerator with degree 2 (10x^2 + 12x + 20). In the numerator on the right you get a sum of two cross products- The first is A(x^2 + 2x + 4). Here A(0 degree) multiplies with a 2nd degree polynomial so the degree is 0 + 2 = 2. The 2nd cross product is (x-2)(Bx+C), a product of two 1st degree polynomials. So this also has a degree of 2.

If you had B instead of Bx+C, you get (x-2)(B) which is a 1st degree polynomial. If you have Bx, you get (x-2)(Bx) which is a 2nd degree polynomial alright, but you are making an unnecessary assumption that this polynomial has no 0 degree terms.

Remember, Bx+C is the standard form of a 1st degree polynomial.

Finally, notice that understanding the degree of polynomials is the key here.

WOW!!!, thank you very much, i got the idea !!!