Question

the trinomial, if possible.

\\5x^2 + 6x - 8\\

not factorable

Explanation:

Identify the coefficients

We begin with the quadratic trinomial:

$$5x^2 + 6x - 8$$

Using Factoring Trinomials, we identify the coefficients:
\(a = 5\), \(b = 6\), and \(c = -8\).

Find key product and sum

Using Finding Factors with Given Sum and Product, we calculate the product \(ac\):

$$ac = 5 \times (-8) = -40$$

We need two integers that multiply to \(-40\) and add to \(b = 6\).

Determine the correct factors

We list pairs of factors of \(-40\) to find the pair that sums to \(6\):

• \(-1 \times 40 = -40\) (sum: \(39\))
• \(-2 \times 20 = -40\) (sum: \(18\))
• \(-4 \times 10 = -40\) (sum: \(6\))

The correct factors are \(-4\) and \(10\).

Rewrite and factor by grouping

Using Trinomial Factoring by Grouping, we split the middle term \(6x\) into \(-4x + 10x\):

$$5x^2 - 4x + 10x - 8$$

Group the terms:

$$(5x^2 - 4x) + (10x - 8)$$

Factor out the greatest common factor from each group:

$$x(5x - 4) + 2(5x - 4)$$

Factor out the common binomial

We factor out the common binomial term \((5x - 4)\):

$$(5x - 4)(x + 2)$$

Answer:

\((5x - 4)(x + 2)\)