How do you factor \[12{{x}^{2}}+69x+45\]?

Hint: In order to find the solution to this question, we will start solving it by taking out common terms like 3, and then we will make necessary calculations to get our equation in simpler quadratic form. Then we will use middle term splitting to simplify further.

Complete step by step answer:
According to the question, we have been asked to find the factors of \[12{{x}^{2}}+69x+45\].
To factorize an equation, one should always first look for the common factors of the coefficients. If the coefficients of the variables don’t have any common factor, then one can proceed with different factoring methods. Here, one can easily see that \[3\] is a common factor of 12, 69, and 45.
Therefore, we will take 3 commons from each term of expression. Therefore, we get
\[12{{x}^{2}}+69x+45=3(4{{x}^{2}}+23x+15)\]
Now, we will use splitting the middle term method to express the obtained quadratic expression as the product of two linear expressions.
We know that when we split the middle term, we split them in such a manner that their algebraic sum is equal to the middle term and their algebraic product is equal to the product first and the last term.
Therefore, we can write the middle term of \[4{{x}^{2}}+23x+15\] as \[23x=20x+3x\]. Hence, we can write the expression as
\[\Rightarrow 3\left( 4{{x}^{2}}+23x+15 \right)=3\left( 4{{x}^{2}}+20x+3x+15 \right)\]
\[\Rightarrow 12{{x}^{2}}+69x+45=3\left( 4{{x}^{2}}+20x+3x+15 \right)\]
Now, we will take 4x common from the first two terms of the obtained expression and then we will take 3 common from next two terms. Hence, we get
\[\Rightarrow 12{{x}^{2}}+69x+45=3\left( 4x\left( x+5 \right)+3\left( x+5 \right) \right)\]
Now, we will take \[\left( x+5 \right)\] common from both the terms. Hence, we get
\[\therefore 12{{x}^{2}}+69x+45=3\left( x+5 \right)\left( 4x+3 \right)\]
Therefore, we can say the factors of \[12{{x}^{2}}+69x+45\] are 3, (x+5), and (4x+3).

Note:
One can use several other methods for factoring the equations. Some of these include the discriminant method or the perfect square method. One should keep in mind that some of these methods might not be able to give proper factorization. Also, we have to be very careful while splitting the terms, because we made a calculation mistake, we might end up with the wrong answer and we may lose our marks.