Question

1. what is the area of the l - shaped carpet below?

Explanation:

Step1: Divide the L - shaped carpet into two rectangles

We can divide the L - shaped carpet into two rectangles. One rectangle has dimensions \((2x + 1)\) and \(3x\), and the other has dimensions \((2x+1)\) and \(x\).

Step2: Calculate the area of the first rectangle

The area formula for a rectangle is \(A = length\times width\). For the first rectangle with length \(l=(2x + 1)\) and width \(w = 3x\), we use the distributive property \((a + b)c=ac+bc\). So, \(A_1=(2x + 1)\times3x=2x\times3x+1\times3x=6x^{2}+3x\).

Step3: Calculate the area of the second rectangle

For the second rectangle with length \(l=(2x + 1)\) and width \(w = x\), using the distributive property \((a + b)c=ac+bc\), we get \(A_2=(2x + 1)\times x=2x\times x+1\times x=2x^{2}+x\).

Step4: Calculate the total area

The total area \(A=A_1 + A_2\). Substitute \(A_1 = 6x^{2}+3x\) and \(A_2=2x^{2}+x\) into the formula: \(A=(6x^{2}+3x)+(2x^{2}+x)\). Combine like terms: \(A=(6x^{2}+2x^{2})+(3x + x)=8x^{2}+4x\).

Answer:

\(8x^{2}+4x\)