Question

completely factor the trinomial, if possible.

\\6x^2 + 22x + 12\\

Explanation:

Factor out the greatest common factor

We begin by finding the greatest common factor (GCF) of the terms in the trinomial \(6x^2 + 22x + 12\).
Using the Factoring Polynomials by GCF knowledge point, we find the GCF of the coefficients \(6\), \(22\), and \(12\), which is \(2\).
Factoring out \(2\) gives:

$$6x^2 + 22x + 12 = 2(3x^2 + 11x + 6)$$

Find factors for grouping

Next, we factor the remaining quadratic trinomial \(3x^2 + 11x + 6\).
Using the Factoring Trinomials knowledge point, we look for two numbers that multiply to \(a \cdot c = 3 \cdot 6 = 18\) and add up to \(b = 11\).
These two numbers are \(9\) and \(2\), since:

$$9 \cdot 2 = 18$$

$$9 + 2 = 11$$

Rewrite and group terms

Using the Trinomial Factoring by Grouping knowledge point, we split the middle term \(11x\) into \(9x + 2x\):

$$3x^2 + 11x + 6 = 3x^2 + 9x + 2x + 6$$

Now, we group the terms into two pairs:

$$(3x^2 + 9x) + (2x + 6)$$

Factor by grouping

We factor out the GCF from each group:

• From \(3x^2 + 9x\), we factor out \(3x\): \(3x(x + 3)\)
• From \(2x + 6\), we factor out \(2\): \(2(x + 3)\)

This gives:

$$3x(x + 3) + 2(x + 3)$$

Since \((x + 3)\) is a common binomial factor, we factor it out:

$$(3x + 2)(x + 3)$$

Write the complete factorization

Finally, we combine the GCF factored out in the first step with the factored trinomial:

$$2(3x + 2)(x + 3)$$

Answer:

\(2(3x + 2)(x + 3)\)