Compare (2b+5)² and 4b²+25: Perfect Square Expansion Problem

To solve this problem, we'll follow these steps:

• Step 1: Expand $(2b + 5)^2$ using the square of a binomial formula.
• Step 2: Compare the expanded form with $4b^2 + 25$.
• Step 3: Determine the relationship between the two expressions based on $b$.

Now, let's work through each step:

Step 1: We begin by expanding $(2b + 5)^2$. Using the formula for the square of a sum, we have:

$(2b + 5)^2 = (2b)^2 + 2 \cdot 2b \cdot 5 + 5^2$

$= 4b^2 + 20b + 25$

Step 2: We now compare this expression to $4b^2 + 25$:

$4b^2 + 20b + 25$ versus $4b^2 + 25$

Step 3: Since $4b^2 + 25$ is shared by both expressions, the comparison depends entirely on the $20b$ term:

If $b = 0$, both expressions are equal.

If $b > 0$, $(2b + 5)^2$ is greater than $4b^2 + 25$ because $20b > 0$.

If $b < 0$, $(2b + 5)^2$ is less than $4b^2 + 25$ because $20b < 0$.

Therefore, the relationship between the two expressions depends on the value of $b$.

The correct choice is: Depends on the value of b.