Square of sum: Identify the greater value

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.

Let's work through determining the relationship between the expressions by expanding and comparing them.

First, expand the right-hand expression $2(x+3)^2$:

• Start with the binomial expansion: $(x+3)^2 = x^2 + 6x + 9$.
• Multiply through by 2: $2(x^2 + 6x + 9) = 2x^2 + 12x + 18$.

Now, consider the left-hand expression $x(2x+3)$:

• Distribute $x$: $x \cdot 2x + x \cdot 3 = 2x^2 + 3x$.

We now have the expanded expressions:

• Left: $2x^2 + 3x$
• Right: $2x^2 + 12x + 18$

To compare these, subtract the left expression from the right:

$(2x^2 + 12x + 18) - (2x^2 + 3x) = 9x + 18$

The result $9x + 18$ shows that for any positive $x$, $9x + 18$ is greater than 0 since both terms are positive. Hence, the right side is always larger than the left side.

Therefore, $x(2x+3) < 2(x+3)^2$ for $x > 0$.

The correct inequality is $<$.

To solve this problem, we'll start by expanding both expressions to compare them.

First, expand the left expression:

$1 + (5 + x)^2 + 3x^2$

Using the formula for expanding a square, $(5 + x)^2$ becomes:

$5^2 + 2 \cdot 5 \cdot x + x^2 = 25 + 10x + x^2$

Thus, the left-hand expression becomes:

$1 + 25 + 10x + x^2 + 3x^2 = 26 + 10x + 4x^2$

Next, simplify the right-hand side:

$5x^2 + 10x + 25$

Now, compare both simplified expressions:

• Left: $26 + 10x + 4x^2$
• Right: $5x^2 + 10x + 25$

Subtract the right expression from the left to see which is greater:

$(26 + 10x + 4x^2) - (5x^2 + 10x + 25)$

$= 26 + 10x + 4x^2 - 5x^2 - 10x - 25$

$= 1 - x^2$

Since $0 < x < 1$, it follows that $-1 < -x^2 < 0$, making $1 - x^2$ positive.

Therefore, the expression on the left is greater than the expression on the right:

$1 + (5 + x)^2 + 3x^2 \gt 5x^2 + 10x + 25$.

To solve this problem, we aim to relate the expression $m(m+n)^2$ and $(2m+n)^2$ given the equation $m^3 + 2m^2n = 4m^2 + n^2$.

The key step is to understand that due to the given constraint equation, without loss of generality or additional specific values for $m$ and $n$, a direct comparison to determine the sign $?$ is not feasible under the constraint $m^3 + 2m^2n = 4m^2 + n^2$. The nature of the relationship defined by the constraint tells us that the relationship between these expressions might hold under special conditions, but projecting one side as definitively greater, less, or equal is not straightforward.

Given this complexity and the lack of further simplifiable information directly provided by the equation, it is not possible to calculate or visually confirm which sign should be appropriately used without more specific values or further simplification details.

Therefore, the answer is: It is not possible to calculate.

The task is to determine the inequality $a(a-b)^2$ compared to $a(a+b)^2 - ab^2$ given $0 < a, b$. Here's how to solve it:

First, let's expand and simplify each expression:

Starting with $a(a-b)^2$, expand as follows:

• $(a-b)^2 = a^2 - 2ab + b^2$.
• Multiplying by $a$ gives: $a(a^2 - 2ab + b^2) = a^3 - 2a^2b + ab^2$.

Next, expand $a(a+b)^2-ab^2$:

• $(a+b)^2 = a^2 + 2ab + b^2$.
• Multiplying by $a$ gives: $a(a^2 + 2ab + b^2) = a^3 + 2a^2b + ab^2$.
• Subtracting $ab^2$ yields: $a^3 + 2a^2b + ab^2 - ab^2 = a^3 + 2a^2b$.

Now compare the two expressions:

• $a(a-b)^2 = a^3 - 2a^2b + ab^2$.
• $a(a+b)^2 - ab^2 = a^3 + 2a^2b$.
• The inequality $a(a-b)^2 ? a(a+b)^2 - ab^2$ is simplified to:\)
• $a^3 - 2a^2b + ab^2 < a^3 + 2a^2b$.
• Cancel $a^3$ from both sides: $-2a^2b + ab^2 < 2a^2b$.
• Bring like terms together: $-4a^2b + ab^2 < 0$.
• Factor $b$: $b(-4a^2 + ab) < 0$.

Given $b > 0$, divide through by $b$:

$-4a^2 + ab < 0$, or equivalently $-4a^2 < -ab$, hence $4a > b$.

Therefore, $a > \frac{b}{4}$.

The inequality holds as $a(a-b)^2 < a(a+b)^2 - ab^2$ when $a > \frac{b}{4}$.

Therefore, the correct choice is:

$<$ When $a > \frac{b}{4}$