a) State or imply the form $\frac{A}{{1 + x}} + \frac{{Bx + C}}{{1 + 2{x^2}}}$

Use any relevant method to evaluate a constant

Obtain one of $A =  - 1$, $B = 2$, $C = 1$

Obtain a second value

Obtain the third value

b) Use correct method to obtain the first two terms of the expansion of ${\left( {1 + x} \right)^{ - 1}}$ or ${\left( {1 + 2{x^2}} \right)^{ - 1}}$

Obtain correct expansion of each partial fraction as far as necessary

Multiply out fully by $Bx + C$, where $BC\,\,\rho \,0$

Obtain answer $3x - 3{x^2} - 3{x^3}$

[Symbolic binomial coefficients, e.g., $\left( \begin{gathered}
   - 1 \hfill \\
  \,\,\,1 \hfill \\ 
\end{gathered}  \right)$ are not sufficient for the first M1. The f.t. is on $A$, $B$, $C$.]

[If $B$ or $C$ omitted from the form of fractions, give B0M1A0A0A0 in (a); M1A1√A1√ in (b), max 4/10.]

[If a constant $D$ is added to the correct form, give M1A1A1A1 and B1 if and only if $D = 0$ is stated.]

[If an extra term $D/\left( {1 + 2{x^2}} \right)$ is added, give B1M1A1A1, and A1 if $C + D = 1$ is resolved to $1/\left( {1 + 2{x^2}} \right)$.]

[In the case of an attempt to expand $3x{\left( {1 + x} \right)^{ - 1}}{\left( {1 + 2{x^2}} \right)^{ - 1}}$, give M1A1A1 for the expansions up to the term in ${x^2}$, M1 for multiplying out fully, and A1 for the final answer.]

[For the identity $3x \equiv \left( {1 + x + 2{x^2} + 2{x^3}} \right)\left( {\alpha  + bx + c{x^2} + d{x^3}} \right)$ give M1A1; then M1A1 for using a relevant method to find two of $\alpha  = 0$, $b = 3$, $c =  - 3$ and $d =  - 3$; and then A1 for the final answer in series form.]