Simplify the given expression, \[\left( 1.5x-4y \right)\left( 1.5x+4y+3 \right)-4.5x+12y\] .

Hint: Modify the given equation as \[\left( 1.5x-4y \right)\left( 1.5x+4y+3 \right)-\left( 4.5x-12y \right)\]. It has two parts \[\left( 1.5x-4y \right)\left( 1.5x+4y+3 \right)\] and \[4.5x-12y\]. Now, take 3 as common in the second part. Finally, take the term \[\left( 1.5x-4y \right)\] as common from the whole and simplify it further.

Complete step-by-step solution
According to the question, we are given an expression and we are asked to simplify it further.
Let us modify the given expression, \[\left( 1.5x-4y \right)\left( 1.5x+4y+3 \right)-4.5x+12y\] .
On modifying the given expression, we get
\[\Rightarrow \left( 1.5x-4y \right)\left( 1.5x+4y+3 \right)-\left( 4.5x-12y \right)\] ……………………………………………….(1)
We can observe that the above expression has two parts.
Let us consider the part before the minus sign as the first part and the part after the minus sign as the second part ………………………………………….(2)
The first part \[\Rightarrow \left( 1.5x-4y \right)\left( 1.5x+4y+3 \right)\] ……………………………………………(3)
The second part \[\Rightarrow 4.5x-12y\] ………………………………..(4)
In the second part, we have the summation of \[4.5x\] and \[12y\]. Also, both are divisible by 3. So, in the second part, we can take 3 as common from the whole.
Now, in equation (3), on taking 3 as common, we get
\[\Rightarrow 3\left( 1.5x-4y \right)\] ………………………………………………………..(5)
From equation (2), equation (3), and equation (5), we get
\[\Rightarrow \left( 1.5x-4y \right)\left( 1.5x+4y+3 \right)-3\left( 1.5x-4y \right)\]
We can observe that the term \[\left( 1.5x-4y \right)\] can be taken as common form the whole.
Now, on taking the term \[\left( 1.5x-4y \right)\] as common from the whole, we get
\[\begin{align}
  & \Rightarrow \left( 1.5x-4y \right)\left\{ \left( 1.5x+4y+3 \right)-3 \right\} \\
 & \Rightarrow \left( 1.5x-4y \right)\left\{ 1.5x+4y+3-3 \right\} \\
\end{align}\]
\[\Rightarrow \left( 1.5x-4y \right)\left( 1.5x+4y \right)\] …………………………………………..(6)
From equation (6), we have the most simplified form of the given expression.
Therefore, the simplified form of the expression is \[\left( 1.5x-4y \right)\left( 1.5x+4y \right)\] .

Note: Whenever this type of question appears where we have to simplify an expression. Always approach this type of question by observing the expression and figure out the part that can be taken as common form the whole. This approach will reduce the time that we normally waste in expanding the expression.