Here are some hints!

For 2.14:

Whenever you are confused by a problem, it is good to start looking at a few specific cases.

• Let's first look at some with $N=2$:  what can we say about the equations $x^2 = 2$, $x^2 = 4$, $x^2 = 0$, or $x^2 = -4$?
  
• We could still look at more examples with $N=4:  what can we say about the equations $x^4 = 2$, $x^4 = 4$, $x^4 = 0$, or $x^4 = -4$?

In general, looking at a few different examples, trying out varying range of values. For example, here we are told $K$ is a real number, but it might be positive, negative, rational, irrational, etc.

For 2.15:

The key here is the Factor Theorem/Fact, but don't be scared by this, it's easier than you think! Consider any question we solve, for example $x^2+3x+2 = 0$ factors as $(x+2)(x+1) = 0$ so we know $x = -2$ or $x = -1$. Do you notice a pattern between the factored form:  $(x+2)\times (x+1)$ and our final solutions $-2$ and $-1$?

For 2.16, 2.20, and 2.26:

Keep in mind our patterns here, for cubes $(a+b)^3$, difference of two squares or difference of two cubes, and sum of two cubes.

• Your first step in 2.16 is good, can you then use one of the factoring formulas/patterns to take the next step? 
  
• For 2.20, keep in mind we can write $x^6 - 1 = (x^2)^3-1^3$ but also $x^6 - 1 = (x^3)^2 - 1^2$.
• For 2.26, you have the right idea, but just be careful how you're using the formula.

For 2.29:

Try to group as $(x^3 + 3x^2) + (2x+6)$. The second group $2x+6$ has a common factor of $2$ so can be written as $2(x+3)$. Can you do something similar for the first group?

For 2.30:

You can try to do a similar thing to Reverse FOIL here, try $xy - 3x+4y+12 = (x + \underline{\ \ })(y + \underline{\ \ })$.