Simplify the algebraic expression $\left( 1.5x-4y \right)\left( 1.5x+4y+3 \right)-4.5x+12y$.

Hint: We complete the multiplication first. We first multiply every term of $\left( 1.5x+4y+3 \right)$ with $1.5x$ and then with $-4y$. After multiplication we are only left with only addition and subtraction. The simplification gives the final solution.

Complete step-by-step solution:
We have to find the simplified form of $\left( 1.5x-4y \right)\left( 1.5x+4y+3 \right)-4.5x+12y$.
The given expression is the binary operations of two variables x and y.
We have one multiplication to complete before completing the addition and subtraction.
In the part of $\left( 1.5x-4y \right)\left( 1.5x+4y+3 \right)$, we first multiply every term of $\left( 1.5x+4y+3 \right)$ with $1.5x$ and then with $-4y$.
Therefore, $1.5x\left( 1.5x+4y+3 \right)=2.25{{x}^{2}}+6xy+4.5x$ and $\left( -4y \right)\left( 1.5x+4y+3 \right)=-6xy-16{{y}^{2}}-12y$.
Now we simplify the whole equation by breaking the brackets.
\[\begin{align}
  & \left( 1.5x-4y \right)\left( 1.5x+4y+3 \right)-4.5x+12y \\
 & =2.25{{x}^{2}}+6xy+4.5x-6xy-16{{y}^{2}}-12y-4.5x+12y \\
\end{align}\]
We have a number of terms all in quadratic power or linear form. We try to find the similar terms and apply the binary operations on their coefficients.
We take the similar terms in parenthesis.
Therefore,
\[\begin{align}
  & 2.25{{x}^{2}}+6xy+4.5x-6xy-16{{y}^{2}}-12y-4.5x+12y \\
 & =2.25{{x}^{2}}-16{{y}^{2}}+\left( 6xy-6xy \right)+\left( 4.5x-4.5x \right)+\left( 12y-12y \right) \\
 & =2.25{{x}^{2}}-16{{y}^{2}} \\
\end{align}\]
Therefore, the terms omit each other.
We get the final answer as $\left( 1.5x-4y \right)\left( 1.5x+4y+3 \right)-4.5x+12y=2.25{{x}^{2}}-16{{y}^{2}}$.

Note: We need to check the variables as coefficients of non-similar terms cannot be simplified.
We can also solve the problem by breaking the multiplication as
$\left( 1.5x-4y \right)\left( 1.5x+4y+3 \right)=\left( 1.5x-4y \right)\left( 1.5x+4y \right)+3\left( 1.5x-4y \right)$
Then we can apply the identity of $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$.