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Here are the solutions to the problems:

1. **8(a)**: \( 4x^2 + 2x + 5 = 4(x + 0.25)^2 + 4.9375 \)

2. **8(b)**: The function \( 4x^2 + 2x + 5 \) **does not** meet the x-axis.

3. **9(a)**: The expression \( 8x^3 - 2x^2 - 8x - 1 \) is already simplified and cannot be factored further into simpler linear and quadratic terms.

4. **9(b)**: The stationary points of the function are at \( x = -\frac{1}{2}, \frac{2}{3} \). At \( x = -\frac{1}{2} \), the function value is \( \frac{3}{2} \).

5. **10(a)**: The smallest value of \( g(x) = x^3 - 5x \) is \( -\frac{10\sqrt{15}}{9} \).

6. **10(b)**: The function \( f(x) \) such that \( f'(x) = \frac{1}{x^3 - 5x} \) is \( f(x) = -\frac{\log(x)}{5} + \frac{\log(x^2 - 5)}{10} \).

7. **11(a)**: The expression \( 2x^2 - 7x \) in the form \( p[(x - q)^2 - r] \) is \( 6.125(0.5714x - 1)^2 - 4.125 \).

8. **11(b)**: The range of the function \( g(x) = 2x^2 - 7x \) is \([18.375, \infty)\).

Do you have any questions or need further details? Here are some related questions:

1. How to complete the square for other quadratic functions?
2. What does the discriminant tell us about the roots of a quadratic equation?
3. How to find and classify stationary points for different functions?
4. What are the implications of a function's derivative on its behavior?
5. How can you determine the range of a quadratic function?

**Tip:** When solving for the stationary points of a cubic function, always evaluate the first derivative and solve for where it equals zero.

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Explore a comprehensive solution to problems involving quadratic and cubic equations, including completing the square, finding stationary points, and determining the range of functions. Suitable for students in Grades 10-12.

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