Are you asking how he knew ? If so, one method is as follows:
The product of the coefficients of the first and last terms is .
and are factors of (the product from the previous step) and sum to (the coefficient of the middle term). So, write the middle term, , as : .
Then, group each pair of terms together: .
Perform any factoring on each group of terms: .
Now, factor : .
Another way to do that (but much more complicated that just factoring directly) is to solve the equation - not by factoring, of course, but by the quadratic formula:
[tex]x= \frac{-5\pm\sqrt{5^2- 4(4)(1)}}{2(4)}= \frac{-5\pm\sqrt{25- 16}}{8}= \frac{-5\pm\sqrt{9}}{8}= \frac{-5\pm3}{8}[tex]
so that and
Now, we can say that [tex]4x^2+ 5x+ 1= 4(x+ \frac{1}{4})(x- 1)= (4x+ 1)(x- 1)[/quote]
(I put the "4" in front to make t he leading coefficient 4.)
That is, as I said, the hard way!
Dear HallsofIvy,
You new avatar make some of the text unreadable. I have seen this in your recent post a number of times and I tried the zoom in/out feature in the browser to eliminate the problem. But it was not helpful. Dont know if this is only a problem with my browser but anyhow I have included a screenshot so that you get an idea about how I see your post.