Let

Now putting,

A(x-1)²+B(x+3)(x-1)+C(x+3)=x²+1

Now put x-1=0

Therefore, x=1

A(0)+B(0)+C(4) =2

Now put x+3=0

Therefore, x=-3

A(-3-1)²+B(0)+C(0) =9+1=10

By equating the coefficient of x², we get, A+B=1

From equation (1), we get,