The area of a rectangle is \( 4 x^{2}-x-5 \) square units. What are the dimensions of the length and width?

To factor the expression \( 4x^2 - x - 5 \), we look for two numbers that multiply to \( 4 \times (-5) = -20 \) and add to \( -1 \) (the coefficient of the middle term). These numbers are \( -5 \) and \( 4 \). Now, we can rewrite the expression: \[ 4x^2 - 5x + 4x - 5 \] Grouping the terms gives: \[ (4x^2 - 5x) + (4x - 5) \] Factoring out the common factors results in: \[ x(4x - 5) + 1(4x - 5) \] Factoring out \( (4x - 5) \), we have: \[ (4x - 5)(x + 1) \] Thus, the dimensions of the rectangle can be considered as \( 4x - 5 \) and \( x + 1 \). In practice, these dimensions imply that there's a relationship between width and length based on the value of \( x \). For example, if \( x = 2 \), the dimensions would be \( 3 \) (width) and \( 13 \) (length), giving an area of 39 square units. This showcases how a simple algebraic expression translates to tangible measurements, bridging the gap between math and the real world!