Writing L-Sentences - Logic Needing some help/guidance in how to write down $L$-sentences that express the following assertions. 
First let $L = {+, *, -, <, 0}$


*

*If a real number has a square root, then its opposite does not.

$\forall x (0<x) \exists y (x*x=y \wedge -(x*x)\neq y)$. Not sure you can use the  $\neq$ though. 


*There exists a quadratic polynomial with real coefficients that has no real root.

I'm pretty stuck on this one. Guess: $\forall x,y,z \exists w (ax^2 + bx + cx = w) \wedge w$


*Every positive integer is the sum of four squares.

$\forall x (x = a^2 + b^2 + c^2 + d^2)$. I do not think I can use squares ($\cdot^2$) though since it is not in the language. So maybe: $\forall x(x = a*a + b*b + c*c + d*d)$


*There are infinitely many positive even integers that can written as the difference of two squares. 

$\forall x \exists y ((0<2x) \wedge y*y= (y-w)(y+w)$ I do not think this is correct.

 A: In the first two formulas, quantifiers are assumed to range over the reals. In the next two, they range over the integers. Note that there are many ways one can write something using our language. 
The formulas below are on the fussy side. For example, there are lots of parentheses. They are technically necessary, but some could be omitted for the sake of readability. In the second question, so many parentheses were necessary that we gave up.
First question: The following sentence works:
$$\forall x\left(\exists y(y\ast y=x)\longrightarrow \lnot(\exists z (z\ast z=(0-x)))                             \right).$$
The above assertion is of course false, in the case $x=0$. If we wanted to write down a true sentence, we would modify the sentence suitably, by insisting that $x\ne 0$.  
There is a little technical thing, I used $0-x$ instead of the natural $-x$, since presumably $-$ is a binary operator. If we want to get rid of "minus" entirely, we can, but since it is a formal part of the language you were given, we might as well use it.
Second question: We could say: 
$$\exists a\exists b\exists c\exists d\exists e\exists f\forall x\forall y\left(\lnot(a\ast (x\ast x)+b\ast(x\ast y)+c\ast(y\ast y)+d\ast x+e\ast y+f=0)\right)$$
Third question: 
$$\forall x\left((0\lt x)\longrightarrow \exists a\exists b\exists c\exists d(x=a\ast a+(b\ast b+(c\ast c+d\ast d)))                 \right).$$
Fourth question: Here, I will give a guide, so you can put it together yourself. To say there are infinitely $y$ with a certain property, we can say that for all $x$, there is a $y$ such that $x\lt y$ and $y$ has the property.
The property is now that $y+y$ is a difference of two squares. That is not difficult to say, particularly since we have $-$ as a a presumably binary operator. 
