The length of a cuboid is three times its height, and the breadth is twice the height. If the height is $(x^2+2)$ cm, find the lateral surface area of the cuboid at x = 2.

The correct option is 360 cm².

Let the height of the cuboid be represented by $h$, the length by $l$, and the breadth by $b$.
According to the problem description:
The height is given by:
$h=x^2+2$
The length $l$ is three times the height:
$l=3h$
The breadth $b$ is twice the height:
$b=2h$

We are asked to find the lateral surface area of the cuboid at $x=2$.
First, let us calculate the value of the height $h$ at $x=2$:
$h={\left(2\right)}^2+2$
$h=4+2=6\text{cm}$

Now, we can find the dimensions of the cuboid at $x=2$ using the calculated height $h=6\text{cm}$:
Length, $l=3\times 6=18\text{cm}$
Breadth, $b=2\times 6=12\text{cm}$
Height, $h=6\text{cm}$

The formula for the lateral surface area (LSA) of a cuboid is:
$\text{LSA}=2h(l+b)$

Substituting the values of $l$, $b$, and $h$ into the formula:
$\text{LSA}=2\times 6\times (18+12)$
$\text{LSA}=12\times 30$
$\text{LSA}=360{\text{cm}}^2$

Therefore, the lateral surface area of the cuboid at $x=2$ is indeed 360 cm².