Factor the polynomial completely. Do NOT put any spaces in your answer!Must show work/steps on your work page𝑥2+5𝑥−24=

To factor the polynomial $x^2 + 5x - 24$ completely, follow these steps:

1. Identify the coefficients: The polynomial is in the form $ax^2 + bx + c$, where $a = 1$, $b = 5$, and $c = -24$.

2. Find two numbers that multiply to $ac$ and add to $b$:

   • Here, $a \cdot c = 1 \cdot (-24) = -24$.
   • We need two numbers that multiply to -24 and add to 5.

3. Determine the pair of numbers:

   • The numbers 8 and -3 work because $8 \cdot (-3) = -24$ and $8 + (-3) = 5$.

4. Rewrite the middle term using these numbers:

   • Rewrite $5x$ as $8x - 3x$.
   • The polynomial becomes $x^2 + 8x - 3x - 24$.

5. Factor by grouping:

   • Group the terms: $(x^2 + 8x) + (-3x - 24)$.
   • Factor out the greatest common factor (GCF) from each group:
     • From $x^2 + 8x$, factor out $x$: $x(x + 8)$.
     • From $-3x - 24$, factor out $-3$: $-3(x + 8)$.

6. Factor out the common binomial factor:

   • The expression is now $x(x + 8) - 3(x + 8)$.
   • Factor out $(x + 8)$: $(x + 8)(x - 3)$.

So, the completely factored form of the polynomial $x^2 + 5x - 24$ is:

$(x+8)(x-3)$