(10)
\[
\frac{x+2}{(x+1)(x^2+1)}
= \frac{A}{x+1} + \frac{Bx + C}{x^2+1}
\]
Clear denominators and equate coefficients.
\[
x+2 = A(x^2+1) + (Bx+C)(x+1)
\]
\[
x+2 = (A+B)x^2 + (B+C)x + (A+C)
\]
\[
\begin{cases}
A + B = 0,\\
B + C = 1,\\
A + C = 2
\end{cases}
\Longrightarrow
A=\tfrac12,\; B=-\tfrac12,\; C=\tfrac32
\]
\[
\boxed{\frac{x+2}{(x+1)(x^2+1)}
= \frac{\tfrac12}{x+1} + \frac{-\tfrac12 x + \tfrac32}{x^2+1}}
\]
Split: simple + irreducible quadratic
(11)
\[
\frac{7x^2-25x+6}{(x^2-2x-1)(3x-2)}
= \frac{Ax+B}{x^2-2x-1} + \frac{C}{3x-2}
\]
Clear denominators, expand, and compare coefficients.
\[
7x^2-25x+6 = (Ax+B)(3x-2) + C(x^2-2x-1)
\]
\[
7x^2-25x+6 = (3A+C)x^2 + (-2A+3B-2C)x + (-2B-C)
\]
\[
\begin{cases}
3A + C = 7,\\
-2A + 3B - 2C = -25,\\
-2B - C = 6
\end{cases}
\Longrightarrow
A=1,\; B=-5,\; C=4
\]
\[
\boxed{\frac{7x^2-25x+6}{(x^2-2x-1)(3x-2)}
= \frac{x-5}{x^2-2x-1} + \frac{4}{3x-2}}
\]
Split: quadratic factor + linear factor
(12)
\[
\frac{x^2+x+1}{x^2+2x+1}
= A + \frac{B}{x+1} + \frac{C}{(x+1)^2}
\quad(\text{since }x^2+2x+1=(x+1)^2)
\]
Clear denominators and match coefficients.
\[
x^2+x+1 = A(x+1)^2 + B(x+1) + C
\]
\[
x^2+x+1 = A x^2 + (2A+B)x + (A+B+C)
\]
\[
\begin{cases}
A=1,\\
2A+B=1,\\
A+B+C=1
\end{cases}
\Longrightarrow
A=1,\; B=-1,\; C=1
\]
\[
\boxed{\frac{x^2+x+1}{(x+1)^2}
= 1 - \frac{1}{x+1} + \frac{1}{(x+1)^2}}
\]
Split: repeated linear factor