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

Use a relevant method to determine a constant

Obtain one of the values $A =  - 2$, $B = 1$, $C = 4$

Obtain a second value

Obtain the third value

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

Obtain correct unsimplified expansion up to the term in ${x^3}$ of each partial fraction Multiply out fully by $Bx + C$, where $BC \ne 0$

Obtain final answer $\frac{5}{2}x - 3{x^2} + \frac{7}{4}{x^3}$, or equivalent

[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.]

[In the case of an attempt to expand $\left( {5x - {x^2}} \right){\left( {1 + x} \right)^{ - 1}}{\left( {2 + {x^2}} \right)^{ - 1}}$, give M1A1A1 for the expansions, M1 for the multiplying out fully, and A1 for the final answer.]

[Allow use of Maclaurin, giving M1A1√A1√ for differentiating and obtaining $f\left( 0 \right) = 0$ and $f'\left( 0 \right) = \frac{5}{2}$, A1√ for $f''\left( 0 \right) =  - 6$, and A1 for $f'''\left( 0 \right) = \frac{{21}}{2}$ and the final answer (the f.t. is on $A$, $B$, $C$ if used).]

[For the identity $5x - {x^2} \equiv \left( {2 + 2x + {x^2} + {x^3}} \right)\left( {\alpha  + bx + c{x^2} + d{x^3}} \right)$ give M1A1; then M1A1 for using a relevant method to obtain two of $\alpha  = 0$, $b = \frac{5}{2}$, $c =  - 3$ and $d = \frac{7}{4}$; then A1 for the final answer in series form.]