Question

4x^2 + 8x - 5

Answer 1

It looks like you've provided a quadratic expression:

\[ 4x^2 + 8x - 5. \]

If you would like to factor it, find its roots, or do something else with it, please specify! Here are a few common operations you might be interested in:

### 1. Factoring:

We can try to factor the quadratic expression:

To factor \( 4x^2 + 8x - 5 \), we look for two numbers that multiply to \( ac = 4 \cdot (-5) = -20 \) and add to \( b = 8 \).

The numbers that work are \( 10 \) and \( -2 \). So, we can rewrite the expression:

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

Then, factor by grouping:

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

So, the factored form is:

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

### 2. Finding Roots:

You could also find the roots of the quadratic equation \( 4x^2 + 8x - 5 = 0 \) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
\]

where \( a = 4 \), \( b = 8 \), and \( c = -5 \).

Plugging in these values:

\[
x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 4 \cdot (-5)}}{2 \cdot 4},
\]
\[
x = \frac{-8 \pm \sqrt{64 + 80}}{8},
\]
\[
x = \frac{-8 \pm \sqrt{144}}{8},
\]
\[
x = \frac{-8 \pm 12}{8}.
\]

This gives two possible solutions:

1. \( x = \frac{4}{8} = \frac{1}{2} \)
2. \( x = \frac{-20}{8} = -\frac{5}{2} \)

### Summary:

- The factored form is \((2x - 1)(2x + 5)\).
- The roots are \( x = \frac{1}{2} \) and \( x = -\frac{5}{2} \).

If you need further assistance or a different calculation, please let me know!