Let

Now putting,

A(x²+1)+(Bx+C)(x+2) = x²+x+1

Ax²+A+Bx²+Cx+2Bx+2C = x²+x+1

(A+B)x²+(C+2B)x+(A+2C) = x²+x+1

Equating coefficients A+B=1…….(i)

A+2C=1

A=1-2C……(ii)

2B+C=1

2B=1-C

2-4C+1-C=2

3-5C=2

-5C=-1

And