Questions: Remember that perimeter means distance around. Write the perimeter of the rectangle to the right as a simplified polynomial. Then factor the polynomial. Write the perimeter of the rectangle as a simplified polynomial. The simplified polynomial for the perimeter is (Do not factor.)

Remember that perimeter means distance around. Write the perimeter of the rectangle to the right as a simplified polynomial. Then factor the polynomial. Write the perimeter of the rectangle as a simplified polynomial. The simplified polynomial for the perimeter is (Do not factor.)
Transcript text: Remember that perimeter means distance around. Write the perimeter of the rectangle to the right as a simplified polynomial. Then factor the polynomial. Write the perimeter of the rectangle as a simplified polynomial. The simplified polynomial for the perimeter is $\square$ (Do not factor.)
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Solution

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Solution Steps

Step 1: Identify the expressions for the sides of the rectangle

The given expressions for the sides of the rectangle are:

  • Length: \(3x + 32\)
  • Width: \(x^2 + 9x\)
Step 2: Write the formula for the perimeter of a rectangle

The formula for the perimeter \(P\) of a rectangle is: \[ P = 2 \times (\text{Length} + \text{Width}) \]

Step 3: Substitute the given expressions into the perimeter formula

Substitute the expressions for the length and width into the formula: \[ P = 2 \times ((3x + 32) + (x^2 + 9x)) \]

Step 4: Simplify the expression inside the parentheses

Combine like terms inside the parentheses: \[ (3x + 32) + (x^2 + 9x) = x^2 + 3x + 9x + 32 = x^2 + 12x + 32 \]

Step 5: Multiply by 2 to find the perimeter

Multiply the simplified expression by 2: \[ P = 2 \times (x^2 + 12x + 32) = 2x^2 + 24x + 64 \]

Final Answer

The simplified polynomial for the perimeter is: \[ 2x^2 + 24x + 64 \]

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