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

Assume \(\frac{17x^2 - 7x + 16}{(2 + 3x^2)(2 - x)} = \frac{Ax + B}{2 + 3x^2} + \frac{C}{2 - x}\).

Multiply through by the denominator: \(17x^2 - 7x + 16 = (Ax + B)(2 - x) + C(2 + 3x^2)\).

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

\(A = -2, B = 3, C = 5\).

Thus, \(f(x) = \frac{-2x + 3}{2 + 3x^2} + \frac{5}{2 - x}\).

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

Use the binomial expansion for \((2 - x)^{-1}\) and \((2 + 3x^2)^{-1}\):

\((2 - x)^{-1} = \frac{1}{2} + \frac{x}{4} + \frac{x^2}{8} + \frac{x^3}{16}\).

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

Substitute back into the partial fractions and expand:

\(\frac{-2x + 3}{2 + 3x^2} = \left(-2x + 3\right)\left(\frac{1}{2} - \frac{3x^2}{4}\right)\).

\(\frac{5}{2 - x} = 5\left(\frac{1}{2} + \frac{x}{4} + \frac{x^2}{8} + \frac{x^3}{16}\right)\).

Combine and simplify to get:

\(4 + \frac{1}{4}x - \frac{13}{8}x^2 + \frac{29}{16}x^3\).

(c) The expansion is valid for \(|x| < \sqrt[3]{\frac{2}{3}}\).