How do you factor the expression \[2{{x}^{2}}+19x+24\]?

Hint: Apply the middle term split method to factorize \[2{{x}^{2}}+19x+24\]. Split 19x into two terms in such a way that their sum is 19x and the product is \[48{{x}^{2}}\]. For this process, find the prime factors of 48 and combine them in such a way so that the conditions are satisfied. Finally, take the common terms together and write \[2{{x}^{2}}+19x+24\] as the product of two terms.

Complete step by step solution:
Here, we have been asked to factorize the quadratic polynomial \[2{{x}^{2}}+19x+24\].
Let us use the middle term split method for the factorization. It states that we have to split middle term which is 19x into two terms such that their sum is 19x and the product is equal to the product of constant term (24) and \[2{{x}^{2}}\], i.e. \[48{{x}^{2}}\]. To do this, first we need to find all the prime factors of 48. So, let us find.
We know that 48 can be written as: - \[48=2\times 2\times 2\times 2\times 3\] as the product of its primes. Now, we have to group these factors such that our conditions of the middle terms split method are satisfied. So, we have,
(i) \[\left( 16x \right)+\left( 3x \right)=19x\]
(ii) \[\left( 16x \right)\times \left( 3x \right)=48{{x}^{2}}\]
Hence, both the conditions of the middle term split method are satisfied. So, the quadratic polynomial can be written as: -
\[\begin{align}
  & \Rightarrow 2{{x}^{2}}+19x+24=2{{x}^{2}}+16x+3x+24 \\
 & \Rightarrow 2{{x}^{2}}+19x+24=2x\left( x+8 \right)+3\left( x+8 \right) \\
\end{align}\]
Taking (x + 8) common in the R.H.S., we get,
\[\Rightarrow 2{{x}^{2}}+19x+24=\left( x+8 \right)\left( 2x+3 \right)\]

Hence, \[\left( x+8 \right)\left( 2x+3 \right)\] is the factored form of the given quadratic polynomial.

Note: One may note that there can be many combinations of grouping the factors of 48 but we need to select the suitable combination, i.e. the combination that satisfies the two conditions of the middle term split method. We also have two different approaches by which we can factorize the quadratic polynomial, they are: completing the square method and the discriminant method. But these methods are used when the factors are complex or irrational.