Answer:
D) x - 3
Step-by-step explanation:
Given rational expression:
[tex]\dfrac{x^2-4}{x^2+x-6} \cdot \dfrac{x^2-9}{x+2}[/tex]
To multiply and simplify the rational expression, begin by factoring both numerators and the denominator of the first fraction.
Both numerators are the difference of two squares.
[tex]\boxed{\begin{array}{c}\underline{\textsf{Difference of Two Squares Formula}}\\\\a^2-b^2=\left(a+b\right)\left(a-b\right)\end{array}}[/tex]
Therefore, the numerator of the first fraction can be rewritten as:
[tex]x^2-4=x^2-2^2\\\\x^2-4=(x+2)(x-2)[/tex]
Similarly, the numerator of the second fraction can be rewritten as:
[tex]x^2-9=x^2-3^2\\\\x^2-9=(x+3)(x-3)[/tex]
Factor the denominator of the first fraction:
[tex]x^2+x-6\\\\x^2+3x-2x-6\\\\x(x+3)-2(x+3)\\\\(x-2)(x+3)[/tex]
Now, rewrite the original expression with the factored numerators and denominator:
[tex]\dfrac{(x+2)(x-2)}{(x-2)(x+3)} \cdot \dfrac{(x+3)(x-3)}{x+2}[/tex]
Multiply:
[tex]\dfrac{(x+2)(x-2)(x+3)(x-3)}{(x-2)(x+3)(x+2)}[/tex]
Cancel the common factors (x - 2), (x + 2) and (x + 3):
[tex]x-3[/tex]
Therefore, the given rational expression simplifies to:
[tex]\huge\boxed{\boxed{x-3}}[/tex]
