Question

The area of a rectangular rug is represented is 8x2 + 10x – 3. What are the dimensions of the rug?

Group of answer choices

(2x + 3)(4x + 1)

(2x – 3)(4x + 4)

(4x – 1)(2x + 3)

(2x – 1)(x + 4)

Answer 1

To find the dimensions of the rectangular rug represented by the area \( 8x^2 + 10x - 3 \), we need to factor the quadratic expression.

1. **Identify \( a \), \( b \), and \( c \)** in \( ax^2 + bx + c \):
- \( a = 8 \)
- \( b = 10 \)
- \( c = -3 \)

2. **Factor the quadratic expression**:
We look for two numbers that multiply to \( ac = 8 \times (-3) = -24 \) and add to \( b = 10 \).

The pair of numbers that works is \( 12 \) and \( -2 \) because:
- \( 12 \times (-2) = -24 \)
- \( 12 + (-2) = 10 \)

3. **Rewrite the expression**:
\[
8x^2 + 12x - 2x - 3
\]
Grouping the terms:
\[
(8x^2 + 12x) + (-2x - 3)
\]
Factor by grouping:
\[
4x(2x + 3) - 1(2x + 3)
\]
This can be factored further:
\[
(4x - 1)(2x + 3)
\]

Thus the area \( 8x^2 + 10x - 3 \) can be factored as \( (4x - 1)(2x + 3) \).

The dimensions of the rug are:

\(
(4x - 1) \text{ and } (2x + 3)
\)

From the given answer choices, the correct one is:

\[
(4x - 1)(2x + 3)
\]