The image contains several quadratic and other polynomial equations. I will solve a few of these for you step by step. If you have specific ones you want solved or if you'd like the solutions for all of them, let me know.

1. $x^2 - 64 = 0$

This is a simple difference of squares: $x^2 - 64 = 0 \implies (x - 8)(x + 8) = 0$ So, $x = 8$ or $x = -8$.

2. $x^2 - 144 = 0$

Another difference of squares: $x^2 - 144 = 0 \implies (x - 12)(x + 12) = 0$ So, $x = 12$ or $x = -12$.

3. $x^2 + 25 = 0$

This has no real solutions because the square of a real number cannot be negative: $x^2 + 25 = 0 \implies x^2 = -25$ Therefore, the solutions are imaginary: $x = 5i$ or $x = -5i$ (where $i$ is the imaginary unit).

4. $3x - 18x^2 = 0$

Factor out the common term $x$: $3x - 18x^2 = 0 \implies x(3 - 18x) = 0$ So, $x = 0$ or $18x = 3 \implies x = \frac{1}{6}$.

5. $2x^2 - 9 = 0$

Solve for $x^2$: $2x^2 - 9 = 0 \implies 2x^2 = 9 \implies x^2 = \frac{9}{2}$ So, $x = \pm \sqrt{\frac{9}{2}} = \pm \frac{3}{\sqrt{2}}$.

These are the first five equations. Let me know if you want the rest or any specific ones solved.

Would you like details on the methods used here or explanations on any step?

Here are five related questions to expand your knowledge:

1. What is the difference of squares, and how do you factor it?
2. How do you handle quadratic equations that have no real solutions?
3. What is the importance of factoring in solving polynomial equations?
4. How do imaginary numbers arise from quadratic equations?
5. How can you check if a solution to a quadratic equation is correct?

Tip: Always look for common factors first when solving polynomial equations; it simplifies the problem significantly.