Hello, 
Can you PLEASE help me?? I cannot understand how to do this! I 
really do appreciate all your help!! 
Find two integers c (positive or negative) for which each 
polynomial can be factored.

x² + x + c

For emphasis I will put a 1 before the x:

x² + 1x + c  

That would factor if c were such that we could think of two
integers which have product c and sum 1. 

So, for example, we think of two integers which have sum 1, 
say +6 and -5, and then if c were their product, -30, it 
would factor.  So choose c as -30.

Checking: 

x² + 1x - 30 factors as
(x + 6)(x - 5)

We could have picked instead two other integers 
whose sum is +1.  For instance we could have picked
-1 and +2, then c = (-1)(+2) = -2.

So another answer is c = -2. Checking,

 x² + x - 2 factors as
(x - 1)(x + 2) 

There are infinitely many answers for c.

----------------------------

x² - 2x + c

We do it the exact same way.

That would factor if c were such that we could think of two
integers which have product c and sum -2.  So we think of two
integers which have sum -2, say -8 and +6, and then if c were
their product, -48, it would factor.  So choose c as -48.

Checking: 

x² - 2x - 48 factors as
(x - 8)(x + 6)

We could have picked instead two other integers 
whose sum is -2.  For instance we could have picked
+5 and -7, then c = (+5)(-7) = -35.

So another answer is c = -35. Checking,

 x² + x - 35 factors as
(x + 5)(x - 7) 

As in all such problems, there are infinitely many 
answers for c.

===============================================

x² - 3x + c

I'll leave that one for you to do. Just think of two
numbers that have sum -3, say -2 and -1, or -8 and +5, 
or -53 and +50, or -4 and +1, or -103 and 100, 
or -999999999 and +999999996, J or .......
Then multiply them together to find a value for c.

Edwin