Question

the area of the rectangle below is $x^2 + 8x + 15$ square meters. one side is $x + 3$ meter.
from which of the following expressions can the length of the other side be determined?
rectangle image with length labeled, one side labeled $x + 3$, area labeled $x^2 + 8x + 15$
length input box meter

Explanation:

Step1: Recall the area formula of a rectangle

The area of a rectangle is given by \( A = \text{length} \times \text{width} \). So, if we know the area and one of the sides (length or width), we can find the other side by dividing the area by the known side. Here, the area \( A = x^{2}+8x + 15 \) and the width (or one side) is \( x + 3 \). So, the length \( l=\frac{x^{2}+8x + 15}{x + 3} \).

Step2: Factor the quadratic expression

We factor the numerator \( x^{2}+8x + 15 \). We need two numbers that multiply to \( 15 \) and add up to \( 8 \). The numbers are \( 3 \) and \( 5 \). So, \( x^{2}+8x + 15=(x + 3)(x + 5) \).

Step3: Simplify the fraction

Now, substitute the factored form into the expression for length: \( l=\frac{(x + 3)(x + 5)}{x + 3} \). Since \( x
eq - 3 \) (because if \( x=-3 \), the width would be \( 0 \), which is not possible for a rectangle), we can cancel out the common factor \( x + 3 \) from the numerator and the denominator. So, \( l=x + 5 \).

Answer:

The length of the rectangle can be determined from the expression \( \frac{x^{2}+8x + 15}{x + 3} \) (which simplifies to \( x + 5 \) meters).