Factorize the following expression
\[{{m}^{2}}-4m-21\]

Hint: To solve this given question, we will use the splitting middle term technique. We will split -4 as the sum or subtraction of two numbers such that the product of those two numbers is -21. After using this we can easily factorize the given expression.

Complete step-by-step solution:
To factorize the given term we will use the method of splitting by middle term, it is given as If an expression is of the form \[a{{x}^{2}}+bx+c=0,\] then the splitting by middle term means splitting b in the form of p and q such that p + q = b and pq = ac, where p and q are integers, they can be positive as well as negative.
So, we are given the expression as \[{{m}^{2}}-4m-21.\] We will take the LCM of 21.
\[\begin{align}
  & 7\left| \!{\underline {\,
  21 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }1 \\
\end{align}\]
So, we have two possible numbers as \[7\times 3=21\]
Now, we have to consider the sign of numbers. If we require -21, then either 7 or 3 must be negative. To reach a conclusion, let us consider the case of addition/subtraction such that we get -4.
And we will get that - 7 + 3 = - 4.
Therefore, the required numbers are -7 and 3.
By using the splitting by middle term in \[{{m}^{2}}-4m-21.\]
\[\Rightarrow {{m}^{2}}-4m-21\]
\[\Rightarrow {{m}^{2}}-7m+3m-21\]
Taking m common, we have,
\[\Rightarrow m\left( m-7 \right)+3\left( m-7 \right)\]
\[\Rightarrow \left( m+3 \right)\left( m-7 \right)\]
So, the factorization of \[{{m}^{2}}-4m-21\] is (m + 3) (m – 7).

Note: A key point to note here on this question is that we have 7 – 3 = 4 but we took – 7 + 3 = – 4. As we had ‘ – 4m’ in the middle term of the above expression. So, we can always shift between a – b = c or b – a = – c to get the result according to the question. This is the main area of mistake in such questions.