Decompose into partial fractions : (8x + 14) / (x + 1)(x + 5)

Given the fraction (8x+14)/
(x+1)(x+5)

We need to rewrite into the form A/(x+1) +
B/(x+5)

==> (8x+14)/ (x+1)(x+5) = A/(x+1) +
B/(x+5)

We will rewrite with the common
denominator.

==> (8x+14)/(x+1)(x+5) = A(x+5)+
B(x+1)]/ (x+1)(x+5)

Now we will multiply by (x+1)(x+5) so
the denominator cancels.

==> 8x + 14 = A(x+5) +
B(x+1)

==> 8x + 14 = Ax + Bx + 5A +
B

==> 8x + 14 = (A+B)x +
(5A+B)

Now we will compare similar
terms.

==> A+B =
8............(1)

==> 5A+B = 14
........(2)

Now we will solve the
system.

We will subtract (1) from
(2).

==> 4A =
6

==> A = 6/4 =
3/2

==> A = 3/2

But
A+B= 8

==> B = 8- A = 8 - 3/2 =
13/2

==> B = 13/2

Now
we will substitute:

==> A/(x+1) + B(x+5) =
(3/2)/(x+1) +
(13/2)/(x+5)

                                   = 3/2(x+2) 
+ 13/2(x+5)

==> (8x+14)/(x+1)(x+5) =
3/2(x+1)  + 13/2(x+5)