Answer:
[tex]\dfrac{12}{x-7},[/tex]
[tex]x\neq 7,\ x\neq -3.[/tex]
Step-by-step explanation:
Consider the expression
[tex]\dfrac{12x+36}{x^2-4x-21}.[/tex]
1. The numerator can be factored as
[tex]12x+36=12(x+3).[/tex]
2. The denominator [tex]x^2-4x-21[/tex] has the discriminant
[tex]D=(-4)^2-4\cdot (-21)\cdot 1=16+84=100.[/tex]
Then
[tex]x_{1,2}=\dfrac{-(-4)\pm \sqrt{100}}{2\cdot 1}=\dfrac{4\pm 10}{2}=7,\ -3.[/tex]
Then
[tex]x^2-4x-21=(x-7)(x+3).[/tex]
Note that restrictions on the variable x are [tex]x\neq 7,\ x\neq -3.[/tex]
3. Simplify the fraction:
[tex]\dfrac{12x+36}{x^2-4x-21}=\dfrac{12(x+3)}{(x-7)(x+3)}=\dfrac{12}{x-7}.[/tex]
