Investigating the product of (x+4)(x+3). We can draw a rectangle of length (x+4) and width (x+3) as shown in the diagram below:

The image demonstrates how to expand the product of two binomials, $(x+4)(x+3)$, by representing it as the area of a large rectangle.

Step 1: Identify the dimensions of the large rectangle and its sub-rectangles. The large rectangle ACEG has a total length of $(x+4)$ and a total width of $(x+3)$. It is divided into four smaller shapes: • Rectangle ABJH: length $x$, width $x$ • Rectangle BCDJ: length $4$, width $x$ • Rectangle HJFG: length $x$, width $3$ • Rectangle JDEF: length $4$, width $3$

Step 2: Calculate the area of each smaller shape. The area of a rectangle is given by length $\times$ width. • Area of ABJH: $x \times x = x^2$ • Area of BCDJ: $4 \times x = 4x$ • Area of HJFG: $x \times 3 = 3x$ • Area of JDEF: $4 \times 3 = 12$

Step 3: Calculate the total area of the large rectangle by summing the areas of the smaller shapes. The total area of ACEG is the sum of the areas of ABJH, BCDJ, HJFG, and JDEF. $Area of ACEG = x^2 + 4x + 3x + 12$ $Area of ACEG = x^2 + (4x + 3x) + 12$ $Area of ACEG = x^2 + 7x + 12$

Step 4: Equate the total area calculated from the sum of parts to the total area calculated from the overall dimensions. The total area of the large rectangle ACEG can also be found by multiplying its total length $(x+4)$ by its total width $(x+3)$. $Area of ACEG = (x+4)(x+3)$ By equating the two expressions for the area of ACEG, we get: $(x+4)(x+3) = x^2 + 7x + 12$ Thus, the product of $(x+4)(x+3)$ is $\boxed{x^2 + 7x + 12}$.

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