Question

the area of the rectangle below is equal to $x^{2}+4x + 3$.
which expression shows the width $w$?
$x - 1$
$x - 2$
$x + 2$
$x + 1$

Explanation:

Step1: Recall the area formula for a rectangle

The area of a rectangle is \(A = \text{length}\times\text{width}\). Here, \(A=x^{2}+4x + 3\) and length \(l=x + 3\). Let width \(w\). So, \(x^{2}+4x + 3=(x + 3)\times w\).

Step2: Factor the quadratic expression

Factor \(x^{2}+4x + 3\). We know that \(x^{2}+4x + 3=x^{2}+x+3x + 3=x(x + 1)+3(x + 1)=(x + 1)(x + 3)\).

Step3: Solve for \(w\)

Since \((x + 3)(x + 1)=(x + 3)\times w\), divide both sides by \(x + 3\) (assuming \(x
eq - 3\)). Then \(w=\frac{(x + 3)(x + 1)}{x + 3}=x + 1\).

Answer:

\(x + 1\)