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 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

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: $4 \times 3 = 12$

The total area of ACEG is the sum of the areas of ABJH, BCDJ, HJFG, and JDEF.

$$\text{Area of ACEG} = x^2 + 4x + 3x + 12$$ $$\text{Area of ACEG} = x^2 + (4x + 3x) + 12$$ $$\text{Area of ACEG} = x^2 + 7x + 12$$

The total area of the large rectangle ACEG can also be found by multiplying its total length $(x+4)$ by its total width $(x+3)$.

$$\text{Area of ACEG} = (x+4)(x+3)$$

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 $\boxed{x^2 + 7x + 12}$.