We are given a rectangle with length $(x+7)$ cm and width $(x+2)$ cm. The area of the rectangle is given as 66 cm$^2$. We need to form a quadratic equation based on this information.

1. Problem Description

We are given a rectangle with length $(x+7)$ cm and width $(x+2)$ cm. The area of the rectangle is given as 66 cm$^2$. We need to form a quadratic equation based on this information.

2. Solution Steps

The area of a rectangle is given by the formula:

$Area = Length \times Width$

In this case, we have:

$Area = (x+7)(x+2)$

We are given that the area is 66 cm$^2$, so we can set up the equation:

$(x+7)(x+2) = 66$

Expanding the left side of the equation, we get:

$x^2 + 2x + 7x + 14 = 66$

$x^2 + 9x + 14 = 66$

Subtract 66 from both sides to set the equation to zero:

$x^2 + 9x + 14 - 66 = 0$

$x^2 + 9x - 52 = 0$

3. Final Answer

The quadratic equation is $x^2 + 9x - 52 = 0$.