Factorize: - \[{{x}^{2}}+7x+10=0\]

Hint: Use the middle term split method to factorize \[{{x}^{2}}+7x+10\]. Split 7x into two terms in such a way that their sum is 7x and the product is \[10{{x}^{2}}\]. For this process, find the prime factors of 10 and combine them in such a manner so that we can get our condition satisfied. Finally, take the common terms together and write \[{{x}^{2}}+7x+10\] as a product of two terms given as (x – a) (x – b). Here, ‘a’ and ‘b’ are called zeroes of the polynomial.

Complete step by step answer:
Here, we have been asked to factorize the quadratic polynomial: \[{{x}^{2}}+7x+10\].
Let us use the middle term split method for the factorization. It says that we have to split the middle term which is 7x into two terms such that their sum is 7x and the product is \[10{{x}^{2}}\]. To do this, first, we need to find all the prime factors of 10. So, let us find it.
We know that 10 can be written as: - 10 = 2 \[\times \] 5 as the product of its primes. Now, we have to group 2 and 5 such that our condition of the middle term split method is satisfied. So, we have,
(i) 5x + 2x = 7x
(ii) 2x \[\times \] 2x = \[10{{x}^{2}}\]
Hence, both the conditions of the middle term split method are satisfied. So, the quadratic polynomial can be written as: -
\[\Rightarrow {{x}^{2}}+7x+10={{x}^{2}}+5x+2x+10\]
Grouping the terms together we have,
\[\begin{align}
  & \Rightarrow {{x}^{2}}+7x+10=x\left( x+5 \right)+2\left( x+5 \right) \\
 & \Rightarrow {{x}^{2}}+7x+10=\left( x+5 \right)\left( x+2 \right) \\
\end{align}\]
Hence, \[\left( x+5 \right)\left( x+2 \right)\] is the factorized form of the given quadratic polynomial.

Note:
One may note that we can use another method for the factorization. The Discriminant method can also be applied to solve the question. What we will do is we will find the solution to the quadratic equation using the discriminant method. The values of x obtained will be assumed as x = a and x = b. Finally, we will consider the product (x – a) (x – b) to get the factorized form.