Question

completely factor the trinomial, if possible.

\\3x^2 + 4x - 15\\

Explanation:

Identify coefficients

We analyze the given quadratic trinomial:

$$3x^2 + 4x - 15$$

Using the Factoring Trinomials concept, we identify the coefficients:

$$a = 3,\quad b = 4,\quad c = -15$$

Find key factors

We calculate the product \(ac\):

$$ac = 3 \times (-15) = -45$$

We need two integers that multiply to \(-45\) and add to \(b = 4\).
Testing factor pairs of \(-45\):

$$9 \times (-5) = -45$$

$$9 + (-5) = 4$$

The required integers are \(9\) and \(-5\).

Rewrite middle term

Using the Trinomial Factoring by Grouping concept, we rewrite the middle term \(4x\) as \(9x - 5x\):

$$3x^2 + 9x - 5x - 15$$

Factor by grouping

We group the terms into two pairs:

$$(3x^2 + 9x) - (5x + 15)$$

Factor out the greatest common factor from each group:

$$3x(x + 3) - 5(x + 3)$$

Extract common binomial

We factor out the common binomial factor \((x + 3)\):

$$(3x - 5)(x + 3)$$

Answer:

\((3x - 5)(x + 3)\)