Find and correct errors of the following mathematical expression:
${\left( {2x} \right)^2} + 5x = 4x + 5x = 9x$

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 law of exponent which is given by ${\left( {ab} \right)^m} = {a^m}{b^m}$.

Complete step by step answer:
In this problem, the mathematical expression ${\left( {2x} \right)^2} + 5x = 4x + 5x = 9x \cdots \cdots \left( 1 \right)$ 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 ${\left( {2x} \right)^2} + 5x$. Now we will use the law of exponent in the first tem of LHS.
The law of exponent is given by ${\left( {ab} \right)^m} = {a^m}{b^m}$.
Let us compare the term ${\left( {2x} \right)^2}$ with ${\left( {ab} \right)^m}$. Therefore, we can say that $a = 2,b = x$ and $m = 2$. So, by using the law ${\left( {ab} \right)^m} = {a^m}{b^m}$, we can write ${\left( {2x} \right)^2} = \left( {{2^2}} \right)\left( {{x^2}} \right) = 4{x^2}$.
Let us simplify the LHS term of the equation $\left( 1 \right)$.
So, we get LHS $ = {\left( {2x} \right)^2} + 5x = 4{x^2} + 5x$.
From the equation $\left( 1 \right)$, we can say that RHS term is $9x$. Now we have the LHS term is $4{x^2} + 5x$ and the RHS term is $9x$. So, we can say that LHS is not equal to RHS. If the RHS term of given expression is $4{x^2} + 5x$ then we can say that given expression is true.
Therefore, the correct mathematical expression is ${\left( {2x} \right)^2} + 5x = 4{x^2} + 5x$.

Note: A mathematical expression must be well-defined. It is not necessary that a mathematical expression must include variables. For example, $1 + 9 = 10$ is a mathematical expression and we can see that there is no variable in this expression. Also $4 + 6 = 9$ 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, $x$ is the variable and variable usually denoted by letter.