(a) Since \((x-2)\) is a factor of \(p(x)\), substituting \(x = 2\) into \(p(x)\) gives:

\(8a - 40 + 2b + 8 = 0\)

\(8a + 2b - 32 = 0\)

\(4a + b = 16\) (Equation 1)

Since \((x-2)\) is also a factor of \(p'(x)\), differentiate \(p(x)\):

\(p'(x) = 3ax^2 - 20x + b\)

Substitute \(x = 2\) into \(p'(x)\):

\(12a - 40 + b = 0\)

\(12a + b = 40\) (Equation 2)

Subtract Equation 1 from Equation 2:

\(12a + b - (4a + b) = 40 - 16\)

\(8a = 24\)

\(a = 3\)

Substitute \(a = 3\) into Equation 1:

\(4(3) + b = 16\)

\(12 + b = 16\)

\(b = 4\)

(b) With \(a = 3\) and \(b = 4\), \(p(x) = 3x^3 - 10x^2 + 4x + 8\).

Since \((x-2)\) is a factor, divide \(p(x)\) by \((x-2)\):

\(p(x) = (x-2)(3x^2 - 4x - 4)\)

Factorise \(3x^2 - 4x - 4\):

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

Thus, \(p(x) = (x-2)^2(3x + 2)\)

Final factorisation: \(3(x + \frac{2}{3})(x-2)^2\)

To divide \(8x^3 + 4x^2 + 2x + 7\) by \(4x^2 + 1\), we perform polynomial long division.

1. Divide the leading term \(8x^3\) by \(4x^2\) to get \(2x\).

2. Multiply \(2x\) by \(4x^2 + 1\) to get \(8x^3 + 2x\).

3. Subtract \(8x^3 + 2x\) from \(8x^3 + 4x^2 + 2x + 7\) to get \(4x^2 + 7\).

4. Divide \(4x^2\) by \(4x^2\) to get \(1\).

5. Multiply \(1\) by \(4x^2 + 1\) to get \(4x^2 + 1\).

6. Subtract \(4x^2 + 1\) from \(4x^2 + 7\) to get \(6\).

The quotient is \(2x + 1\) and the remainder is \(6\).

To divide \(2x^4 + 1\) by \(x^2 - x + 2\), we perform polynomial long division.

1. Divide the leading term \(2x^4\) by \(x^2\) to get \(2x^2\).

2. Multiply \(2x^2\) by \(x^2 - x + 2\) to get \(2x^4 - 2x^3 + 4x^2\).

3. Subtract \(2x^4 - 2x^3 + 4x^2\) from \(2x^4 + 1\) to get \(2x^3 - 4x^2 + 1\).

4. Divide \(2x^3\) by \(x^2\) to get \(2x\).

5. Multiply \(2x\) by \(x^2 - x + 2\) to get \(2x^3 - 2x^2 + 4x\).

6. Subtract \(2x^3 - 2x^2 + 4x\) from \(2x^3 - 4x^2 + 1\) to get \(-2x^2 - 4x + 1\).

7. Divide \(-2x^2\) by \(x^2\) to get \(-2\).

8. Multiply \(-2\) by \(x^2 - x + 2\) to get \(-2x^2 + 2x - 4\).

9. Subtract \(-2x^2 + 2x - 4\) from \(-2x^2 - 4x + 1\) to get \(-6x + 5\).

The quotient is \(2x^2 + 2x - 2\) and the remainder is \(-6x + 5\).