How do you factor \[3{{x}^{2}}+27x+60\]?

Hint: First take 3 common from all the terms present. Now, use the middle term split method to factorize the obtained equation. Split the middle term 9x into two terms such that their sum is 9x and the product is \[20{{x}^{2}}\]. For this process, find the prime factors of 20 and combine them in such a way so that we can get our conditions satisfied. Finally, take the common terms together and write \[3{{x}^{2}}+27x+60\] as a product of two terms given as \[\left( x-a \right)\left( x-b \right)\], where ‘a’ and ‘b’ are called zeroes of the polynomial.

Complete step-by-step answer:
Here, we have been asked to factorize the quadratic polynomial: \[3{{x}^{2}}+27x+60\]. Taking 3 common from all the terms, we get,
\[\Rightarrow 3{{x}^{2}}+27x+60=3\left( {{x}^{2}}+9x+20 \right)\]
Now, we need to factorize \[\left( {{x}^{2}}+9x+20 \right)\]. Let us use the middle term split method for the factorization. It states that we have to split the middle term which is 9x into two terms such that their sum is 9x and the product is equal to \[20{{x}^{2}}\]. To do this, first we need to find all the prime factors of 20. So, let us find.
We know that 20 can be written as: - \[20=2\times 2\times 5\] as the product of its primes. Now, we have to group these factors such that our conditions of the middle term split method are satisfied. So, we have,
(i) \[4x+5x=9x\]
(ii) \[4x\times 5x=20{{x}^{2}}\]
Hence, both the conditions of the middle term split method are satisfied.
So, the quadratic polynomial can be written as: -
\[\begin{align}
  & \Rightarrow 3\left( {{x}^{2}}+9x+20 \right)=3\left( {{x}^{2}}+4x+5x+20 \right) \\
 & \Rightarrow 3\left( {{x}^{2}}+9x+20 \right)=3\left[ x\left( x+4 \right)+5\left( x+4 \right) \right] \\
\end{align}\]
Taking \[\left( x+4 \right)\] common in the R.H.S., we get,
\[\Rightarrow 3\left( {{x}^{2}}+9x+20 \right)=3\left( x+4 \right)\left( x+5 \right)\]
Hence, \[3\left( x+4 \right)\left( x+5 \right)\] is the factored 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 of the quadratic equation using discriminant method. The values of x obtained will be assumed as x = a and x = b. Finally, we will consider the product \[\left( x-a \right)\left( x-b \right)\] to get the factored form.