Question

rectangle abcd is shown below. the length of the rectangle can be expressed by $5x + 1$ and the width can be expressed by $x + 4$.
what is an expression that can be used to represent the area of the rectangle?
$5x^{2}+21x + 4$
$12x + 10$
$6x + 5$
$5x^{2}+6x + 4$

Explanation:

Step1: Recall the formula for the area of a rectangle

The area \(A\) of a rectangle is given by \(A=\text{length}\times\text{width}\). Here, length \(l = 5x + 1\) and width \(w=x + 4\). So, \(A=(5x + 1)(x + 4)\).

Step2: Use the distributive property (FOIL method)

First: \(5x\times x=5x^{2}\).
Outer: \(5x\times4 = 20x\).
Inner: \(1\times x=x\).
Last: \(1\times4 = 4\).
Then, \(A=5x^{2}+20x+x + 4\).

Step3: Combine like - terms

Combine \(20x\) and \(x\) (since \(20x+x=(20 + 1)x=21x\)). So, \(A=5x^{2}+21x + 4\).

Answer:

\(5x^{2}+21x + 4\) (corresponding to the first option).