Compare Quadratic Expressions: x(2x+3) ? 2(x+3)² with x > 0

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 $<$.