factor:

4s^2+4s+1

To factor the expression 4s^2 + 4s + 1, we can use the factoring method called "perfect square trinomial."

A perfect square trinomial is a trinomial that can be expressed as the square of a binomial. In this case, we want to determine if the given expression can be factored as the square of a binomial.

Let's start by examining the three terms of the expression. We have 4s^2 as the first term, 4s as the second term, and 1 as the third term.

First, we look at the coefficient of the squared term, which is 4. Taking the square root of this coefficient, we get 2. We can also observe that when we square 2, we get 4.

Next, we check the coefficient of the middle term, which is 4s. We divide it by 2, which gives us 2s.

Finally, we check the constant term, which is 1. We compare it to the square of (2s), which is (2s)^2 = 4s^2.

Since the constant term of 1 is equal to the square of the binomial (2s)^2, we can conclude that the given expression is a perfect square trinomial.

Now, let's express the perfect square trinomial as the square of a binomial:

(2s + 1)^2

Therefore, the factored form of 4s^2 + 4s + 1 is (2s + 1)^2.