(a) To express \(f(x)\) in partial fractions, assume \(f(x) = \frac{A}{1+2x} + \frac{B}{1-2x} + \frac{C}{2+x}\).

Multiply through by the denominator \((1+2x)(1-2x)(2+x)\) to clear the fractions:

\(2 + 11x - 10x^2 = A(1-2x)(2+x) + B(1+2x)(2+x) + C(1+2x)(1-2x)\).

Expand and equate coefficients to solve for \(A, B,\) and \(C\):

\(A = -2, B = 1, C = 4\).

Thus, \(f(x) = \frac{-2}{1+2x} + \frac{1}{1-2x} + \frac{4}{2+x}\).

(b) Expand each partial fraction up to \(x^2\):

\(\frac{-2}{1+2x} = -2(1 - 2x + 4x^2 + \, \ldots) = -2 + 4x - 8x^2\).

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

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

Add the expansions:

\(f(x) = (-2 + 4x - 8x^2) + (1 + 2x + 4x^2) + (2 - x + \frac{x^2}{2})\).

Combine like terms:

\(f(x) = 1 + 5x - \frac{7}{2}x^2\).