I have to factor this equation, which is a perfect square:

2p^2+4p+1

How do I do this?

This expression can be factorized, but not with rational coefficients, nor is it a perfect square.

The easiest way is to find the roots, either by completing the squares, or using the quadratic formula. I chose the latter, which gives as roots
-1 ±√2/2
So the expression factors as
(sqrt(2)p+sqrt(2)+1)*(sqrt(2)p+sqrt(2)-1);
or equivalently,
(p+1+sqrt(2)/2)(2p+2-sqrt(2))

Oh! Of course it isn't a perfect square, because of the 2, right?

Jeez do I ever hate trick questions. Thanks a lot. :) I was really stuck and now I know why! Haha!

- <3

In the quadratic formula, if the term inside the square-root sign, "b²-4ac" is not zero, it will not be a perfect square.

In this case, b²-4ac = 4²-4*2*1 = 8, so you can quickly tell that it won't be a perfect square.
Do check to see if you have copied the question correctly. :)

To factor the equation 2p^2 + 4p + 1, you need to determine if it's a perfect square trinomial. A perfect square trinomial can be factored as the square of a binomial.

To check if the given equation is a perfect square trinomial, compare it to the general form of a perfect square trinomial:

(a + b)^2 = a^2 + 2ab + b^2

In this case, a = p and b = 1.

Now, let's substitute these values into the general form:

(p + 1)^2 = p^2 + 2p(1) + 1

As you can see, the expression matches the general form, which means it is a perfect square trinomial.

Therefore, you can factor 2p^2 + 4p + 1 as (p + 1)^2.