Final answer:
The value of b in the given equation, upon expansion and comparison of like terms, appears to be 2. However, this value is not among the provided options, indicating there might be an error in the question or in the answer choices.
Step-by-step explanation:
To find the value of b in the equation x^2 + 4x + 9 = (x + b)^2 + 5, we need to expand the right-hand side and then equate the coefficients from both sides of the equation. First, let us expand (x + b)^2:
(x + b)^2 = x^2 + 2xb + b^2
Now the equation looks like this:
x^2 + 4x + 9 = x^2 + 2xb + b^2 + 5
Subtract x^2 from both sides:
4x + 9 = 2xb + b^2 + 5
Now, let's equate the coefficients of the like terms:
- For the x terms: 4 = 2b, which gives us b = 2
- For the constant terms: 9 = b^2 + 5, solving for b^2 gives us b^2 = 4
Since b^2 = 4, the value of b can be either 2 or -2. However, since we already found that b = 2 from the coefficient of x, the only consistent solution is b = 2. This is not one of the options provided, which means there might be an error in the provided options or in our calculation. We should review the question and available options for a possible error if this situation arises during a test or homework.