Question

SHOW YOUR WORK.
Factor 8x^2 + 16x - 10 completely. Which of the following expressions is a factor?
a x - 10
b 2x + 3
c 4x - 1
d 2x + 5

Answer 1

To factor the quadratic expression \( 8x^2 + 16x - 10 \) completely, we follow these steps:

1. **Factor out the greatest common factor (GCF)**:
The coefficients of the terms are \( 8 \), \( 16 \), and \( -10 \). The GCF of these numbers is \( 2 \). So we can start by factoring out \( 2 \):

\[
8x^2 + 16x - 10 = 2(4x^2 + 8x - 5)
\]

2. **Factor the quadratic**:
Now we need to factor the quadratic expression \( 4x^2 + 8x - 5 \). We will look for two numbers that multiply to \( ac \) (where \( a = 4 \) and \( c = -5 \)), which is \( 4 \times -5 = -20 \), and add up to \( b = 8 \).

The pairs of factors of \( -20 \) that add up to \( 8 \) are:
- \( 10 \) and \( -2 \)

So we can rewrite \( 4x^2 + 8x - 5 \) using these numbers:

\[
4x^2 + 10x - 2x - 5
\]

Now we can group the terms:

\[
(4x^2 + 10x) + (-2x - 5)
\]

Factor out the common factors in each group:

\[
2x(2x + 5) - 1(2x + 5)
\]

Now we can factor out \( (2x + 5) \):

\[
(2x + 5)(2x - 1)
\]

3. **Combine everything**:
We can now combine this back with the factor of \( 2 \) we factored out earlier:

\[
8x^2 + 16x - 10 = 2(2x + 5)(2x - 1)
\]

The completely factored form of \( 8x^2 + 16x - 10 \) is:

\[
2(2x + 5)(2x - 1)
\]

4. **Check the options**:
Now, we need to see which of the given expressions is a factor of the quadratic expression:

- a: \( x - 10 \) (not a factor)
- b: \( 2x + 3 \) (not a factor)
- c: \( 4x - 1 \) (not in the factorization)
- d: \( 2x + 5 \) (is a factor)

The factor from the original expression is **\((2x + 5)\)**, so the correct answer is option **d**: \( 2x + 5 \).