Answer :
To determine the excluded values of [tex]\( x \)[/tex] for the function [tex]\(\frac{x+4}{-3x^2 + 12x + 36}\)[/tex], we need to find the values of [tex]\( x \)[/tex] that make the denominator equal to zero. Excluded values occur at these points because division by zero is undefined in mathematics.
Let's follow these steps to find the excluded values:
1. Identify the Denominator:
[tex]\[ -3x^2 + 12x + 36 \][/tex]
2. Set the Denominator Equal to Zero:
[tex]\[ -3x^2 + 12x + 36 = 0 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
We need to solve the quadratic equation [tex]\(-3x^2 + 12x + 36 = 0\)[/tex].
The solutions to this quadratic equation are:
[tex]\[ x = -2 \quad \text{and} \quad x = 6. \][/tex]
These values make the denominator zero, thereby causing the fraction to be undefined. Thus, the excluded values of [tex]\( x \)[/tex] are:
[tex]\[ x = -2 \quad \text{and} \quad x = 6. \][/tex]
So, the correct choice from the given options is:
[tex]\[ \boxed{x = -2, x = 6} \][/tex]
Let's follow these steps to find the excluded values:
1. Identify the Denominator:
[tex]\[ -3x^2 + 12x + 36 \][/tex]
2. Set the Denominator Equal to Zero:
[tex]\[ -3x^2 + 12x + 36 = 0 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
We need to solve the quadratic equation [tex]\(-3x^2 + 12x + 36 = 0\)[/tex].
The solutions to this quadratic equation are:
[tex]\[ x = -2 \quad \text{and} \quad x = 6. \][/tex]
These values make the denominator zero, thereby causing the fraction to be undefined. Thus, the excluded values of [tex]\( x \)[/tex] are:
[tex]\[ x = -2 \quad \text{and} \quad x = 6. \][/tex]
So, the correct choice from the given options is:
[tex]\[ \boxed{x = -2, x = 6} \][/tex]