Question 278999
Find the value(s) of {{{k}}} for which {{{k+5x+x^2}}} can be factored.
<pre>
Write it in descending order:

{{{x^2+5x+k}}}

Suppose it factors as

{{{(x+a)(x+b)}}}

Then we can choose a and b as any two integers which
add to 5.

We can choose a=1, b=4, or a=2, b=3 

With the first choice,

{{{(x+a)(x+b)}}} becomes

{{{(x+1)(x+4)}}}

which when FOILed out becomes

{{{x^2+5x+4}}}, so if k=4 then {{{k+5x+x^2}}} can be factored

With the second choice,

{{{(x+a)(x+b)}}} becomes

{{{(x+2)(x+3)}}}

which when FOILed out becomes

{{{x^2+5x+6}}}, so if k=6 then {{{k+5x+x^2}}} can be factored

So the answer is:  If k = 4 or 6, then {{{k+5x+x^2}}} can

be factored.  Otherwise it can't.

Edwin</pre>