Question

AREA The area of a rectangle is $x^2+5x+6$. What binomial expressions represent the side lengths of the rectangle?

Short answer

The length and width of the rectangle are$\left(x+3\right)$ and $\left(x+2\right)$.

Step-by-step solution

Step 1. Write the formula for area of rectangle.

Area of rectangle is given by $A=l\times w$.

Step 2. Find the GCF.

Find the GCF of $x^2,5x$and 6.

role="math" localid="1648027471940" $$\begin{gathered} x^2=1\cdot x\cdot x \\ 5x=1\cdot 5\cdot x \\ 6=1\cdot 2\cdot 3 \end{gathered}$$

Therefore, GCF is 1.

Step 3. Apply factoring pattern of ax2+bx+c.

There are two numbers having product of 6 and sum of $5$. The numbers are $2$and $3$.

$x^2+mx+nx+6=x^2+2x+3x+6$

Step 4. Group terms with common factors and factor out GCF.

Group terms with common factors and factor out GCF from each grouping.

$$\begin{gathered} \left(x^2+2x\right)+\left(3x+6\right)=x\left(x+2\right)+3\left(x+2\right) \\ =\left(x+3\right)\left(x+2\right) \end{gathered}$$

The factors of$x^2+5x+6$ are $\left(x+3\right)\left(x+2\right)$. Therefore, length and width of the rectangle are$\left(x+3\right)$ and $\left(x+2\right)$.