Final answer:
To find the length of the rectangle, substitute the given area and width values into the area formula and solve for the length by factoring or using the quadratic formula.
Step-by-step explanation:
To find the length of the rectangle, we can use the formula for the area of a rectangle:
Area = length x width
Since we are given the area as (3x² + 7x - 6) and the width as (x + 3), we can substitute these values into the formula and solve for the length:
(3x² + 7x - 6) = length x (x + 3)
Next, we can distribute the length:
3x³ + 7x² - 6x = x² + 3x²
Combine like terms:
3x³ + 4x² - 6x = 0
Factor out a factor of x:
x(3x² + 4x - 6) = 0
Set each factor equal to zero:
x = 0 or 3x² + 4x - 6 = 0
Since the width cannot be zero, we can focus on the second equation:
3x² + 4x - 6 = 0
This quadratic equation can be factored, or you can use the quadratic formula to solve for x and then find the length.