Answer:
See below.
Step-by-step explanation:
First, distribute:
[tex]=\frac{1}{x(x+1)}[/tex]
Now, perform partial fraction decomposition. This is only two factors, so we only need linear functions:
[tex]\frac{1}{x(x+1)} =\frac{A}{x}+\frac{B}{x+1}[/tex]
Now, multiply everything by x(x+1):
[tex]1=A(x+1)+B(x)[/tex]
Now, solve for each variable. Let's let x=-1:
[tex]1=A(-1+1)+B(-1)[/tex]
[tex]1=0A-B=-B[/tex]
[tex]B=-1[/tex]
Now, let's let x=0:
[tex]1=A(0+1)+B(0)[/tex]
[tex]A=1[/tex]
So:
[tex]\frac{1}{x(x+1)}=\frac{1}{x}-\frac{1}{(x+1)}[/tex]
Double Check:
[tex]\frac{1}{x}-\frac{1}{(x+1)}=\frac{(x+1)}{x(x+1)}-\frac{x}{x(x+1)}[/tex]
[tex]=\frac{x-x+1}{x(x+1)} =\frac{1}{x^2+x}[/tex]
