Alright, let's break down the given quadratic expression step-by-step.
We are given the quadratic expression [tex]\( 4x^2 + 4x + 1 \)[/tex].
### Step 1: Identify the Coefficients
In a quadratic expression of the form [tex]\( ax^2 + bx + c \)[/tex]:
- [tex]\( a \)[/tex] is the coefficient of [tex]\( x^2 \)[/tex],
- [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex],
- [tex]\( c \)[/tex] is the constant term.
For our expression [tex]\( 4x^2 + 4x + 1 \)[/tex]:
- [tex]\( a = 4 \)[/tex],
- [tex]\( b = 4 \)[/tex],
- [tex]\( c = 1 \)[/tex].
### Step 2: Verify the Standard Form
Our quadratic expression is already in standard form: [tex]\( ax^2 + bx + c \)[/tex].
### Step 3: Checking For Any Special Forms
Here, we should check if the given quadratic expression can be simplified further or factored into simpler forms.
#### Step 3.1: Expand (if Necessary)
The expression is already expanded as [tex]\( 4x^2 + 4x + 1 \)[/tex], which means it's written in its standard quadratic form.
#### Step 3.2: Factorization (If Possible)
We check if the expression can be factored. Let’s consider the expression:
[tex]\[ 4x^2 + 4x + 1 \][/tex]
To factorize this quadratic expression, we can start by checking if it's a perfect square trinomial.
A perfect square trinomial takes the form [tex]\( (ux + v)^2 = u^2x^2 + 2uvx + v^2 \)[/tex].
We see that:
[tex]\[ (2x + 1)^2 = (2x + 1)(2x + 1) = 4x^2 + 4x + 1 \][/tex]
Thus, [tex]\( 4x^2 + 4x + 1 \)[/tex] can be written as:
[tex]\[ 4x^2 + 4x + 1 = (2x + 1)^2 \][/tex]
### Step 4: Conclusions
We verified that:
[tex]\[ 4x^2 + 4x + 1 \][/tex]
is, in fact, a perfect square trinomial and can be written as:
[tex]\[ (2x + 1)^2 \][/tex]
So, the detailed analysis confirms that the quadratic expression [tex]\( 4x^2 + 4x + 1 \)[/tex] is correct and has been confirmed in its expanded form as well as its factored form.
