Question

answer the question in this part. each correct answer will receive 4 credits. clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. utilize the information provided for each question to determine your answer. note that diagrams are not necessarily drawn to scale. for all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.

1. the length of a rectangle is represented by $x^{2}+3x + 2$, and the width is represented by $4x$. express the perimeter of the rectangle as a trinomial. express the area of the rectangle as a trinomial.

Explanation:

Step1: Recall the formula for the perimeter of a rectangle

The formula for the perimeter \(P\) of a rectangle is \(P = 2(l + w)\), where \(l\) is the length and \(w\) is the width. Given \(l=x^{2}+3x + 2\) and \(w = 4x\).
Substitute into the formula: \(P=2((x^{2}+3x + 2)+4x)\)

Step2: Simplify the expression inside the parentheses

\((x^{2}+3x + 2)+4x=x^{2}+(3x + 4x)+2=x^{2}+7x + 2\)

Step3: Multiply by 2

\(P = 2(x^{2}+7x + 2)=2x^{2}+14x + 4\)

Step4: Recall the formula for the area of a rectangle

The formula for the area \(A\) of a rectangle is \(A=l\times w\). Substitute \(l=x^{2}+3x + 2\) and \(w = 4x\)
\(A=(x^{2}+3x + 2)\times4x\)

Step5: Use the distributive property

\(A=4x\times x^{2}+4x\times3x+4x\times2\)
\(A = 4x^{3}+12x^{2}+8x\)

Answer:

The perimeter of the rectangle as a trinomial is \(2x^{2}+14x + 4\). The area of the rectangle as a trinomial is \(4x^{3}+12x^{2}+8x\)