Question: Consider the following rectangle.
$2x$
$5$
$12$
$3x$
How can we express the area of the rectangle in terms of the variable $x$?
The rectangle is partitioned into 4 colored sub-rectangles: top-left yellow, top-right light blue, bottom-left red, and bottom-right green. position_hint = center; the full rectangle has height $2x + 5$ and width $12 + 3x$.
1. **State the problem:** We want to find the area of a rectangle with height $2x + 5$ and width $12 + 3x$ in terms of the variable $x$.
2. **Formula for area of a rectangle:**
$$\text{Area} = \text{height} \times \text{width}$$
3. **Apply the formula:**
$$\text{Area} = (2x + 5)(12 + 3x)$$
4. **Expand the product:**
$$= (2x)(12) + (2x)(3x) + 5(12) + 5(3x)$$
$$= 24x + 6x^2 + 60 + 15x$$
5. **Combine like terms:**
$$6x^2 + (24x + 15x) + 60 = 6x^2 + 39x + 60$$
6. **Final expression:**
The area of the rectangle in terms of $x$ is
$$\boxed{6x^2 + 39x + 60}$$
This expression represents the total area of the rectangle as a quadratic polynomial in $x$.