2019 NJC P1 Q4

Show that \(\frac{p{{x}^{2}}+\left( 4p-q \right)x+\left( 4p+q \right)}{\left( 1-x \right){{\left( 2+x \right)}^{2}}}=\frac{p}{1-x}+\frac{q}{{{\left( 2+x \right)}^{2}}}\).[1]

Find the values of \(p\), \(q\) and \(r\) such that the series expansion of

\(\frac{p{{x}^{2}}+\left( 4p-q \right)x+\left( 4p+q \right)}{\left( 1-x \right){{\left( 2+x \right)}^{2}}}+r\ln \left( 1-3x \right)\)

up to and including the term in \({{x}^{2}}\) is \(11-3x+{{x}^{2}}\).[4]

Find the range of the values of \(x\) for which the expansion is valid.[2]