Question

Which of the following simplifies to the same expression as
$4x^{2}-3x^{2}+5x-24$ ? (1 point)
$2(x-12)$
$(x+8)(x-3)$
$4x(x-6)$
$(x+6)(x-4)$

Answer 1

To solve the problem, we'll simplify the expression \(4x^{2}-3x^{2}+5x-24\) first.

1. Start by combining the like terms of \(4x^2\) and \(-3x^2\):
\[
4x^2 - 3x^2 = x^2
\]
So, the expression simplifies to:
\[
x^2 + 5x - 24
\]

Next, we will factor this quadratic expression \(x^2 + 5x - 24\).

To factor \(x^2 + 5x - 24\), we look for two numbers that multiply to \(-24\) (the constant term) and add to \(5\) (the coefficient of \(x\)). The numbers \(8\) and \(-3\) fit this requirement because:
\[
8 \cdot (-3) = -24 \quad \text{and} \quad 8 + (-3) = 5
\]

Thus, we can factor the quadratic as:
\[
x^2 + 5x - 24 = (x + 8)(x - 3)
\]

Now, let’s compare this result with each of the provided options:

1. **Option 1: \(2(x-12)\)**
- This expands to \(2x - 24\), which does not match \(x^2 + 5x - 24\).

2. **Option 2: \((x+8)(x-3)\)**
- We already found that this matches \(x^2 + 5x - 24\).

3. **Option 3: \(4x(x-6)\)**
- This expands to \(4x^2 - 24x\), which does not match \(x^2 + 5x - 24\).

4. **Option 4: \((x+6)(x-4)\)**
- This expands to \(x^2 - 4x + 6x - 24 = x^2 + 2x - 24\), which does not match \(x^2 + 5x - 24\).

The only option that simplifies to the same expression as \(4x^{2}-3x^{2}+5x-24\) is:

\[
\boxed{(x+8)(x-3)}
\]