Question

completely factor the trinomial, if possible.

\\20x^2 - 37x + 15\\

Explanation:

Identify the coefficients of the trinomial

We begin with the given quadratic trinomial:

$$20x^2 - 37x + 15$$

Using the Factoring Trinomials method, we identify the coefficients:

• \(a = 20\)
• \(b = -37\)
• \(c = 15\)

Find the key product and sum

Using the Finding Factors with Given Sum and Product concept, we calculate:

• Product \(ac = 20 \times 15 = 300\)
• Sum \(b = -37\)

We need to find two integers that multiply to \(300\) and add up to \(-37\). Since the product is positive and the sum is negative, both integers must be negative.
Let's test factor pairs of \(300\):

• \(-10 \times -30 = 300\) (sum is \(-40\))
• \(-12 \times -25 = 300\) (sum is \(-37\))

The two required integers are \(-12\) and \(-25\).

Rewrite the middle term

Using the Trinomial Factoring by Grouping method, we split the middle term \(-37x\) using \(-12x\) and \(-25x\):

$$20x^2 - 12x - 25x + 15$$

Factor by grouping

Group the terms into two pairs:

$$(20x^2 - 12x) - (25x - 15)$$

Factor out the greatest common factor (GCF) from each group:

• From the first group \(20x^2 - 12x\), factor out \(4x\):

$$4x(5x - 3)$$

• From the second group \(-25x + 15\), factor out \(-5\):

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

Combine the factored groups:

$$(4x - 5)(5x - 3)$$

Answer:

\((4x - 5)(5x - 3)\)