When written in factored form, which two polynomials have a factor of x + 12? a. x2 + 10x – 24 b. x2 – 8x – 48 c. x2 – 12x + 27 d. x2 + 15x + 36 e. x2 + 8x + 12

You should write x squared as x^2

both a and d have x + 12 as a factor  Let me know if anything here doesn't make sense.  

Step-by-step explanation:

You should write x squared as x^2 Let's factor each though.  I am going to use the method of finding d and e where d + e = b and d * e = a * c where a, b and c are ax^2 + bx + c in a quadratic polynomial.  To do this first find all factors of a*c and try adding them all.  

a) x^2 + 10x - 24, a = 1, b = 10 and d = -24

a*c = -24 = d*e so let's try out all factors and see what happens when we add them because d + e = 10

-1 and 24 don't work

1 and -24 don't work

2 and -12 don't work

-2 and 12 does work so there's the answer.

x^2 - 2x + 12x - 24

x(x - 2) + 12(x - 2)

(x + 12)(x-2)  And hey, x + 12!

b) x^2 - 8x - 48 a*c = -48 = d*e and d+e = -8

-1 and 48 doens't work

1 and -48 doesn't work

2 and -24 doesn't work

-2 and 24 doesn't work

3 and -16 doesn't work

-3 and 16 doesn't work

4 and -12 DOES work

x^2 + 4x - 12x - 48

x(x + 4) - 12(x + 4)

(x - 12)(x + 4)

No x + 12 so this is not a correct answer

c) x^2 - 12x + 27 a*c = 27 = d*e and d+e = -12

1 and 27 doesn't work

-1 and -27 doesn't work

3 and 9 doesn't work

-3 and -9 does work

x^2 - 12x + 27

x^2 - 3x - 9x + 27

x(x - 3) - 9(x - 3)

(x-9)(x-3) No x+12 so not a right answer.

d) x^2 + 15x + 36 where a*c = 36 = d*e and d+e = 15

1 and 36 doesn't work

-1 and -36 doesn't work

2 and 18 doesn't work

-2 and -18 doesn't work

3 and 12 DOES work

x^2 + 15x + 36

x^2 + 3x + 12x + 36

x(x + 3) + 12(x + 3)

(x+12)(x+3)  And here is x + 12 again so it is one of the right answers.

e) x^2 + 8x + 12 where a*c = 12 = d*e and d+e = 8

1 and 2 doesn't work

-1 and -2 doesn't work

2 and 6 DOES work

x^2 + 8x + 12

x^2 + 2x + 6x + 12

x(x + 2) + 6(x + 2)

(x+6)(x+2) No x+12 so not  an answer.

this means a and d are the right answer for this question.

(a) the volume of the cone is 20/3 in3

hence, option a is correct.