Sure, let's work through this problem step-by-step to find the expression that represents [tex]\( f(x) + g(x) \)[/tex].
### Step 1: Define the Functions
We start by writing the given functions:
[tex]\[ f(x) = x^2 + 2x + 1 \][/tex]
[tex]\[ g(x) = 3(x + 1)^2 \][/tex]
### Step 2: Expand [tex]\( g(x) \)[/tex]
To make it easier to add the two functions, we need to expand [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 3(x + 1)^2 \][/tex]
Let's expand [tex]\( (x + 1)^2 \)[/tex]:
[tex]\[ (x + 1)^2 = x^2 + 2x + 1 \][/tex]
Now, multiply by 3:
[tex]\[ g(x) = 3(x^2 + 2x + 1) \][/tex]
[tex]\[ g(x) = 3x^2 + 6x + 3 \][/tex]
### Step 3: Add [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]
Next, we add the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ f(x) + g(x) = (x^2 + 2x + 1) + (3x^2 + 6x + 3) \][/tex]
Now, combine like terms:
[tex]\[ f(x) + g(x) = x^2 + 3x^2 + 2x + 6x + 1 + 3 \][/tex]
Combine the coefficients:
[tex]\[ f(x) + g(x) = 4x^2 + 8x + 4 \][/tex]
### Conclusion
The expression that represents [tex]\( f(x) + g(x) \)[/tex] is:
[tex]\[ f(x) + g(x) = 4x^2 + 8x + 4 \][/tex]
