The problem asks to find the perimeter of a decagon, given the expressions for two sides: $x^2 + 2x + 40$ and $x^2 - x + 190$. Since it's a decagon, it has 10 sides. We assume it is a regular decagon, which means all the sides have equal length. So we can equate the two given expressions and find the value of x. Then, we use the value of x to calculate the side length and multiply it by 10 to obtain the perimeter of the decagon.

1. Problem Description

The problem asks to find the perimeter of a decagon, given the expressions for two sides: $x^2 + 2x + 40$ and $x^2 - x + 190$. Since it's a decagon, it has 10 sides. We assume it is a regular decagon, which means all the sides have equal length. So we can equate the two given expressions and find the value of x. Then, we use the value of x to calculate the side length and multiply it by 10 to obtain the perimeter of the decagon.

2. Solution Steps

First, equate the expressions for the two sides to find $x$:

$x^2 + 2x + 40 = x^2 - x + 190$

Subtract $x^2$ from both sides:

$2x + 40 = -x + 190$

Add $x$ to both sides:

$3x + 40 = 190$

Subtract $40$ from both sides:

$3x = 150$

Divide both sides by $3$:

$x = 50$

Now, substitute the value of $x$ into either of the expressions to find the side length of the decagon. Let's use $x^2 + 2x + 40$:

$s = (50)^2 + 2(50) + 40$

$s = 2500 + 100 + 40$

$s = 2640$

Finally, the perimeter $P$ of the decagon is 10 times the side length $s$:

$P = 10s$

$P = 10(2640)$

3. Final Answer

$P = 26400$