To find the perimeter of the rectangle, the formula for the perimeter is:

$P = 2 \times (\text{Length} + \text{Width})$

Here, the length is given as $2x^2 + 3x + 1$, and the width is $x + 5$.

Step 1: Sum the Length and Width

$\text{Length + Width} = (2x^2 + 3x + 1) + (x + 5)$

Combine like terms: $\text{Length + Width} = 2x^2 + 4x + 6$

Step 2: Multiply by 2 to Get the Perimeter

$P = 2 \times (2x^2 + 4x + 6)$

Distribute the 2: $P = 4x^2 + 8x + 12$

Final Answer:

The perimeter of the rectangle is: $P = 4x^2 + 8x + 12$

Would you like me to evaluate this further for a specific value of $x$, or provide additional details?

5 Related Questions:

1. What is the perimeter when $x = 0$?
2. Can the area of the rectangle be calculated using these dimensions?
3. How does changing $x$ affect the perimeter of the rectangle?
4. What is the perimeter when $x = 2$?
5. Can you show how the perimeter formula simplifies step-by-step for a specific value of $x$?

Tip:

When working with polynomials, always group like terms carefully to avoid calculation errors!