Find and correct errors of the following mathematical expression:
$\left( {a - 4} \right)\left( {a - 2} \right) = {a^2} - 8$

Hint: In the given problem, to find errors first we will simplify the LHS term and compare it with the RHS term. For this, we will use the identity $\left( {x - p} \right)\left( {x - q} \right) = {x^2} - \left( {p + q} \right)x + pq$.

Complete step by step answer:
In this problem, the mathematical expression $\left( {a - 4} \right)\left( {a - 2} \right) = {a^2} - 8 \cdots \cdots \left( 1 \right)$ is given.
We have to find and correct the errors in this expression. For this, first we will simplify the LHS term. The LHS term of the equation $\left( 1 \right)$ is $\left( {a - 4} \right)\left( {a - 2} \right)$. Now we will use the identity $\left( {x - p} \right)\left( {x - q} \right) = {x^2} - \left( {p + q} \right)x + pq$.
So, by using this identity we can write $\left( {a - 4} \right)\left( {a - 2} \right) = {a^2} - \left( {4 + 2} \right)a + \left( 4 \right)\left( 2 \right)$. Let us simplify this equation. Therefore, we can write $\left( {a - 4} \right)\left( {a - 2} \right) = {a^2} - 6a + 8$.
From the equation $\left( 1 \right)$, we can say that the RHS term is ${a^2} - 8$. Now we have the LHS term is ${a^2} - 6a + 8$ and the RHS term is ${a^2} + 8$. So, we can say that LHS is not equal to RHS. If the RHS term of given expression is ${a^2} - 6a + 8$ then we can say that given expression is true.

Therefore, the correct mathematical expression is $\left( {a - 4} \right)\left( {a - 2} \right) = {a^2} - 6a + 8$.

Note: In some problems, we can use the identity $\left( {x + p} \right)\left( {x + q} \right) = {x^2} + \left( {p + q} \right)x + pq$. A mathematical expression must be well-defined. It is not necessary that a mathematical expression must include variables. For example, $1 - 4 = - 3$ is a mathematical expression and we can see that there is no variable in this expression. Also $1 - 4 = 3$ is mathematical expression but we can say that it is not a valid (correct) mathematical expression as LHS is not equal to RHS. If there are a minimum of two terms and one mathematical operation in the expression then we can say that it is a mathematical expression. In the given problem, $a$ is the variable and variable usually denoted by letter.