What is the quotient in simplest form? State any restrictions on the variable.

Start Fraction x squared minus 4 over x minus 3 End Fraction divided by Start Fraction x plus 2 over x squared plus x minus 12 End Fraction
(1 point)
Responses

Start Fraction left-parenthesis x minus 2 right-parenthesis left-parenthesis x plus 2 right-parenthesis squared over left-parenthesis x minus 3 right-parenthesis squared left-parenthesis x plus 4 right-parenthesis End Fraction, x not equal to negative 4, x not equal to 3
Image with alt text: Start Fraction left-parenthesis x minus 2 right-parenthesis left-parenthesis x plus 2 right-parenthesis squared over left-parenthesis x minus 3 right-parenthesis squared left-parenthesis x plus 4 right-parenthesis End Fraction , Image with alt text: x not equal to negative 4 , Image with alt text: x not equal to 3

Start Fraction left-parenthesis x minus 2 right-parenthesis squared left-parenthesis x plus 4 right-parenthesis over x plus 2 End Fraction, x not equal to negative 2
Image with alt text: Start Fraction left-parenthesis x minus 2 right-parenthesis squared left-parenthesis x plus 4 right-parenthesis over x plus 2 End Fraction , Image with alt text: x not equal to negative 2

x squared plus 2 x minus 8, x not equal to negative 4, x not equal to 3
Image with alt text: x squared plus 2 x minus 8 , Image with alt text: x not equal to negative 4 , Image with alt text: x not equal to 3

x squared plus 2 x minus 8, x not equal to negative 4, x not equal to negative 2, x not equal to 3
Image with alt text: x squared plus 2 x minus 8 , Image with alt text: x not equal to negative 4 , Image with alt text: x not equal to negative 2 , Image with alt text: x not equal to 3
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1 answer

To simplify the expression

\[ \frac{\frac{x^2 - 4}{x - 3}}{\frac{x + 2}{x^2 + x - 12}}, \]

we start by rewriting it as follows:

\[ \frac{x^2 - 4}{x - 3} \cdot \frac{x^2 + x - 12}{x + 2}. \]

Next, we can factor the expressions in the fractions. The numerator \(x^2 - 4\) can be factored as:

\[ x^2 - 4 = (x - 2)(x + 2). \]

The polynomial \(x^2 + x - 12\) can be factored by finding two numbers that multiply to \(-12\) and add to \(1\). These numbers are \(4\) and \(-3\):

\[ x^2 + x - 12 = (x - 3)(x + 4). \]

Now we can substitute these factorizations into the original expression:

\[ \frac{(x - 2)(x + 2)}{x - 3} \cdot \frac{(x - 3)(x + 4)}{x + 2}. \]

Next, we can cancel the common factors. The \((x + 2)\) in the numerator and denominator cancels out, and the \((x - 3)\) in the numerator and denominator also cancels out, leaving us with:

\[ \frac{(x - 2)(x + 4)}{1} = (x - 2)(x + 4). \]

Now, we simplify this to:

\[ x^2 + 2x - 8. \]

Now, we need to state any restrictions on the variable \(x\). The original expressions had the following restrictions:

  • From \(x - 3\) in the denominator, \(x \neq 3\).
  • From \(x + 2\) in the denominator, \(x \neq -2\).
  • The \(x + 2\) from the numerator doesn't impose any further restrictions.

Thus, our final answer is:

\[ x^2 + 2x - 8, \quad x \neq 3, \quad x \neq -2. \]