(i) Express \(f(x)\) in partial fractions:

Assume \(\frac{3x}{(1+x)(1+2x^2)} = \frac{A}{1+x} + \frac{Bx+C}{1+2x^2}\).

Multiply through by \((1+x)(1+2x^2)\) to clear the denominators:

\(3x = A(1+2x^2) + (Bx+C)(1+x)\).

Expand and equate coefficients:

\(3x = A + 2Ax^2 + Bx + Bx^2 + C + Cx\).

Equating coefficients gives:

\(A + C = 0\), \(B + C = 3\), \(2A + B = 0\).

Solving these equations, we find \(A = -1\), \(B = 2\), \(C = 1\).

Thus, \(f(x) = \frac{-1}{1+x} + \frac{2x+1}{1+2x^2}\).

(ii) Expand \(f(x)\) in ascending powers of \(x\):

First, expand \(\frac{-1}{1+x}\) using the geometric series:

\(\frac{-1}{1+x} = -(1 - x + x^2 - x^3 + \cdots)\).

Next, expand \(\frac{2x+1}{1+2x^2}\):

\(\frac{2x+1}{1+2x^2} = (2x+1)(1 - 2x^2 + 4x^4 - \cdots)\).

Multiply out to get terms up to \(x^3\):

\((2x+1)(1 - 2x^2) = 2x + 1 - 4x^3 - 2x^2\).

Combine the expansions:

\(f(x) = -(1 - x + x^2 - x^3) + (2x + 1 - 2x^2 - 4x^3)\).

Simplify to get:

\(f(x) = 3x - 3x^2 - 3x^3\).

(i) Express \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in partial fractions:

Assume \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2} = \frac{A}{1 - 2x} + \frac{B}{2 + x} + \frac{C}{(2 + x)^2}\).

Multiply through by the denominator \((1 - 2x)(2 + x)^2\) to get:

\(4 + 5x - x^2 = A(2 + x)^2 + B(1 - 2x)(2 + x) + C(1 - 2x)\).

Expand and equate coefficients to find \(A = 1\), \(B = 1\), \(C = -2\).

(ii) Expand \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in ascending powers of \(x\):

Use the partial fractions: \(\frac{1}{1 - 2x}\), \(\frac{1}{2 + x}\), and \(\frac{-2}{(2 + x)^2}\).

Expand each term using binomial series:

\(\frac{1}{1 - 2x} = 1 + 2x + 4x^2 + \cdots\)

\(\frac{1}{2 + x} = \frac{1}{2}(1 - \frac{x}{2} + \frac{x^2}{4} + \cdots)\)

\(\frac{-2}{(2 + x)^2} = -\frac{1}{2}(1 - x + x^2 + \cdots)\)

Combine the expansions up to \(x^2\):

\(1 + \frac{9}{4}x + \frac{15}{4}x^2\).

(a) Express \(f(x)\) in partial fractions:

Assume \(\frac{21 - 8x - 2x^2}{(1 + 2x)(3 - x)^2} = \frac{A}{1 + 2x} + \frac{B}{3 - x} + \frac{C}{(3 - x)^2}\).

Find constants \(A, B, C\) by equating coefficients:

\(A = 2, \ B = 2, \ C = -3\).

(b) Expand each partial fraction:

\(\frac{2}{1 + 2x} = 2(1 - 2x + 4x^2 + \ldots)\)

\(\frac{2}{3 - x} = \frac{2}{3}(1 + \frac{x}{3} + \frac{x^2}{9} + \ldots)\)

\(\frac{-3}{(3 - x)^2} = -\frac{1}{3^2}(1 + \frac{2x}{3} + \frac{3x^2}{9} + \ldots)\)

Combine and simplify terms up to \(x^2\):

\(f(x) = \frac{7}{3} - 4x + \frac{215}{27}x^2\).

(i) Express \(\frac{1+x}{(1-x)(2+x^2)}\) in partial fractions:

Assume \(\frac{1+x}{(1-x)(2+x^2)} = \frac{A}{1-x} + \frac{Bx+C}{2+x^2}\).

Multiply through by \((1-x)(2+x^2)\) to get:

\(1+x = A(2+x^2) + (Bx+C)(1-x)\).

Expand and collect terms:

\(1+x = 2A + Ax^2 + Bx - Bx^2 + C - Cx\).

Equate coefficients:

\(A = \frac{2}{3}, \ B = \frac{2}{3}, \ C = \frac{1}{3}\).

(ii) Expand \(\frac{1+x}{(1-x)(2+x^2)}\) in ascending powers of \(x\):

Use the partial fractions: \(\frac{2/3}{1-x} + \frac{(2/3)x + 1/3}{2+x^2}\).

Expand \(\frac{1}{1-x}\) as \(1 + x + x^2 + \ldots\).

Expand \(\frac{1}{2+x^2}\) as \(\frac{1}{2}(1 - \frac{1}{2}x^2 + \ldots)\).

Multiply and combine terms:

\(\frac{2}{3}(1 + x + x^2) + \frac{1}{2}(\frac{2}{3}x - \frac{1}{3})(1 - \frac{1}{2}x^2)\).

Combine and simplify to get:

\(\frac{1}{2} + x + \frac{3}{4}x^2\).

(i) Express \(\frac{5x + 3}{(x + 1)^2(3x + 2)}\) in partial fractions as:

\(\frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{3x+2}\)

Equating and solving for constants, we find \(A = 1, B = 2, C = -3\).

(ii) Expand each term:

\(\frac{1}{(x+1)^2} = (1 + x)^{-2} \approx 1 - 2x + 3x^2\)

\(\frac{1}{3x+2} = \frac{1}{2}(1 - \frac{3}{2}x + (\frac{3}{2}x)^2) \approx \frac{1}{2} - \frac{3}{4}x + \frac{9}{8}x^2\)

Combine and simplify:

\(\frac{5x + 3}{(x + 1)^2(3x + 2)} \approx \frac{3}{2} - \frac{11}{4}x + \frac{29}{8}x^2\)