Let's factor the expression [tex]\( 3r^2 + 6r + 3 \)[/tex] completely, step by step.
### Step 1: Identify Common Factors
First, we observe that each term in the expression [tex]\( 3r^2 + 6r + 3 \)[/tex] has a common factor of 3. Hence, we factor out 3 from the expression:
[tex]\[
3r^2 + 6r + 3 = 3(r^2 + 2r + 1)
\][/tex]
### Step 2: Factor the Quadratic Expression
Next, we need to factor the quadratic expression [tex]\( r^2 + 2r + 1 \)[/tex]. This can be done by looking for two numbers that add up to 2 (the coefficient of [tex]\( r \)[/tex]) and multiply to 1 (the constant term).
The numbers that satisfy both conditions are 1 and 1, so we can write:
[tex]\[
r^2 + 2r + 1 = (r + 1)(r + 1) = (r + 1)^2
\][/tex]
### Step 3: Combine the Factors
Now we put this back into our expression:
[tex]\[
3(r^2 + 2r + 1) = 3(r + 1)^2
\][/tex]
So, the completely factored form of the given expression [tex]\( 3r^2 + 6r + 3 \)[/tex] is:
[tex]\[
3(r + 1)^2
\][/tex]
This is the final factored expression.
