To determine which polynomials have a factor of [tex]\( x+12 \)[/tex], we will factor each polynomial individually and look for [tex]\( x+12 \)[/tex] among the factors.
### 1. [tex]\( x^2 + 15x + 36 \)[/tex]
To factor [tex]\( x^2 + 15x + 36 \)[/tex], we need to find two numbers that multiply to 36 and add to 15. Those numbers are 3 and 12.
[tex]\[
x^2 + 15x + 36 = (x+3)(x+12)
\][/tex]
The factor is [tex]\( x + 12 \)[/tex].
### 2. [tex]\( x^2 - 8x - 48 \)[/tex]
To factor [tex]\( x^2 - 8x - 48 \)[/tex], we need to find two numbers that multiply to -48 and add to -8. Those numbers are -12 and 4.
[tex]\[
x^2 - 8x - 48 = (x-12)(x+4)
\][/tex]
The factor is not [tex]\( x + 12 \)[/tex].
### 3. [tex]\( x^2 + 8x + 12 \)[/tex]
To factor [tex]\( x^2 + 8x + 12 \)[/tex], we need to find two numbers that multiply to 12 and add to 8. Those numbers are 2 and 6.
[tex]\[
x^2 + 8x + 12 = (x+2)(x+6)
\][/tex]
The factor is not [tex]\( x + 12 \)[/tex].
### 4. [tex]\( x^2 - 12x + 27 \)[/tex]
To factor [tex]\( x^2 - 12x + 27 \)[/tex], we need to find two numbers that multiply to 27 and add to -12. Those numbers are -3 and -9.
[tex]\[
x^2 - 12x + 27 = (x-3)(x-9)
\][/tex]
The factor is not [tex]\( x + 12 \)[/tex].
### 5. [tex]\( x^2 + 10x - 24 \)[/tex]
To factor [tex]\( x^2 + 10x - 24 \)[/tex], we need to find two numbers that multiply to -24 and add to 10. Those numbers are 12 and -2.
[tex]\[
x^2 + 10x - 24 = (x+12)(x-2)
\][/tex]
The factor is [tex]\( x + 12 \)[/tex].
### Conclusion
The polynomials [tex]\( x^2 + 15x + 36 \)[/tex] and [tex]\( x^2 + 10x - 24 \)[/tex] have a factor of [tex]\( x + 12 \)[/tex].
