Question

How do you factor \[6{x^2} + 9x + 4x + 6\] ?

Answer 1

Hint: To solve this question, we first need to pair up \[6{x^2}\] with either \[9x\] or \[4x\] , and the remaining terms together. Further, find the common terms, and simplify the expression until there are only two terms remaining.

Complete step by step solution:
The Given Polynomial is a Quadratic Trinomial. Our objective is to factorize this polynomial into simpler terms.
For this, we need to pair the terms which have adjacent powers of the variable. For e.g., here, \[6{x^2}\] and \[9x\] can be said to have adjacent powers of the variable{ \[{x^2}\] and \[x\] vary by 1 in terms of exponential power}. Also, \[4x\] and \[6\] have adjacent powers of \[x\] .
Thus, we can obtain two pairs as:
  \[(6{x^2} + 9x) + (4x + 6)\]
Now for each bracket, we need to take the common factor out of the bracket. For e.g., in \[(6{x^2} + 9x)\] , the common factor that divides both terms inside the brackets is \[3x\] ( \[6{x^2} \div 3x = 2x\] and \[9x \div 3x = 3\] ).
After taking the common factors out of the bracket, we get:
  \[3x(2x + 3) + 2(2x + 3)\]
Now, we also observe that in both the larger terms, i.e., \[3x(2x + 3)\] and \[2(2x + 3)\] , we have a common factor, which is \[(2x + 3)\] . Thus, \[(2x + 3)\] can be taken as common, and we can apply the distributive property to simplify this polynomial further.
On applying distributive property, i.e., \[ab + ac = a(b + c)\] , we get:
  \[3x(2x + 3) + 2(2x + 3) = (3x + 2)(2x + 3)\]
Thus, the final factorization of the polynomial \[6{x^2} + 9x + 4x + 6\] is \[(3x + 2)(2x + 3)\] .
So, the correct answer is “ \[(3x + 2)(2x + 3)\] ”.

Note: In this question, terms need to be paired correctly. It needs to be remembered that to factorize quadratic polynomials, we have to pair up the terms which have adjacent powers of the given variable. Any other combination of pairing will result in an incorrect answer.