Let me start with the most recent questions (I'll still get to the other subjects).

For 2.18, remember you want integer x values where the function is negative. I think you solved for the range where it is positive there.

For 2.26, if L is the length and W is the width, we know LW = 35 and 2L+2W = 24. Try to use these to solve for the side lengths.

For 2.27, the $x_1$ and $x_2$ can sometimes be confusing. If we use $r$ and $s$ for the roots, we know $ax^2 + bx + c = 0$ has roots $r$ and $s$. Note this means that $ar^2 + br + c = 0$ and $as^2+bs+c = 0$. Then we want to solve the equation $a(x+r)^2+b(x+r) + c = 0$. Expand this out. Remember that $x$ is the only variable. What is the constant term of this quadratic (in x)?

For 2.28, this one is a good basic practice with Vieta's theorem. We want to find $(x_1)^2 + (x_2)^2$ where $x_1$ and $x_2$ are the roots. As a hint, how does $(x_1+x_2)^2$ compare to $(x_1)^2 + (x_2)^2$?

For 1.30, start by grouping the first two terms and the second to terms, to get that the expression is equal to $(a^5+b^5)(a^2-b^2)$. There should be factoring formulas for both these terms you can use from here.

Hope this helps!