To determine which expression is equivalent to [tex]\(\frac{x^2+5 x+6}{x^2+7 x+10} \div \frac{x+4}{x-1}\)[/tex], we should follow these steps:
1. Rewrite the Division as Multiplication by the Reciprocal:
Division of fractions is the same as multiplication by the reciprocal. Therefore, we rewrite the given expression as:
[tex]\[
\frac{x^2+5 x+6}{x^2+7 x+10} \div \frac{x+4}{x-1} = \frac{x^2+5 x+6}{x^2+7 x+10} \times \frac{x-1}{x+4}
\][/tex]
2. Multiply the Fractions:
To multiply fractions, we multiply the numerators together and the denominators together:
[tex]\[
\frac{(x^2+5 x+6) \cdot (x-1)}{(x^2+7 x+10) \cdot (x+4)}
\][/tex]
3. Factorize the Expressions:
Before multiplying, it helps to factorize the polynomials to simplify the expression.
- The numerator [tex]\(x^2 + 5x + 6\)[/tex] factors as: [tex]\((x+2)(x+3)\)[/tex]
- The denominator [tex]\(x^2 + 7x + 10\)[/tex] factors as: [tex]\((x+2)(x+5)\)[/tex]
- Now we substitute these factorizations into our expression:
[tex]\[
\frac{(x+2)(x+3) \cdot (x-1)}{(x+2)(x+5) \cdot (x+4)}
\][/tex]
4. Simplify the Expression:
We can cancel out the common factor [tex]\((x+2)\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{(x+3)(x-1)}{(x+5)(x+4)}
\][/tex]
5. Expand Both Numerator and Denominator:
Finally, expand the remaining factors to get a single polynomial expression in the numerator and the denominator:
- Numerator: [tex]\((x+3)(x-1) = x^2 + 2x - 3\)[/tex]
- Denominator: [tex]\((x+5)(x+4) = x^2 + 9x + 20\)[/tex]
Therefore, the resulting simplified expression is:
[tex]\[
\frac{x^2 + 2x - 3}{x^2 + 9x + 20}
\][/tex]
Thus, the expression equivalent to [tex]\(\frac{x^2+5 x+6}{x^2+7 x+10} \div \frac{x+4}{x-1}\)[/tex] is:
Answer: C. [tex]\(\frac{x^2 + 2x - 3}{x^2 + 9x + 20}\)[/tex]
