To determine which of the given expressions could be equivalent to [tex]\( x^2 + bx - 36 \)[/tex] with [tex]\( b \)[/tex] being negative, we need to expand each expression and compare it to the given polynomial form.
### Option A: [tex]\((x+3)(x-12)\)[/tex]
Expand:
[tex]\[ (x + 3)(x - 12) = x^2 - 12x + 3x - 36 = x^2 - 9x - 36 \][/tex]
Here, the polynomial is [tex]\( x^2 - 9x - 36 \)[/tex]. The coefficient of [tex]\( x \)[/tex] is [tex]\(-9\)[/tex], which is negative.
### Option B: [tex]\((x-2)(x+18)\)[/tex]
Expand:
[tex]\[ (x - 2)(x + 18) = x^2 + 18x - 2x - 36 = x^2 + 16x - 36 \][/tex]
Here, the polynomial is [tex]\( x^2 + 16x - 36 \)[/tex]. The coefficient of [tex]\( x \)[/tex] is [tex]\( 16 \)[/tex], which is positive and does not satisfy the condition.
### Option C: [tex]\((x-13)(x-3)\)[/tex]
Expand:
[tex]\[ (x - 13)(x - 3) = x^2 - 3x - 13x + 39 = x^2 - 16x + 39 \][/tex]
Here, the polynomial is [tex]\( x^2 - 16x + 39 \)[/tex]. The constant term is not [tex]\(-36\)[/tex]; this option does not match the requirement.
### Option D: [tex]\((x+4)(x+9)\)[/tex]
Expand:
[tex]\[ (x + 4)(x + 9) = x^2 + 9x + 4x + 36 = x^2 + 13x + 36 \][/tex]
Here, the polynomial is [tex]\( x^2 + 13x + 36 \)[/tex]. The constant term is incorrect as it is positive rather than negative [tex]\(-36\)[/tex].
### Option E: [tex]\((x-9)(x+4)\)[/tex]
Expand:
[tex]\[ (x - 9)(x + 4) = x^2 + 4x - 9x - 36 = x^2 - 5x - 36 \][/tex]
Here, the polynomial is [tex]\( x^2 - 5x - 36 \)[/tex]. The coefficient of [tex]\( x \)[/tex] is [tex]\(-5\)[/tex], which is negative.
Only the expressions in Option A and Option E have a negative coefficient for the [tex]\( x \)[/tex]-term and a constant term of [tex]\(-36\)[/tex].
Given all the expanded forms, the correct answer is:
[tex]\[ A. (x + 3)(x - 12) \][/tex]
[tex]\[ E. (x - 9)(x + 4) \][/tex]
