Find the Missing Symbol: (5+x)² + 3x² ? 5x² + 10x + 25

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:

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