Certainly! Let's go through this problem step-by-step to fully understand the process of simplifying the given expression.
The problem involves dividing and then multiplying rational expressions. Let’s go through it step-by-step:
1. Given Expression:
[tex]\[\frac{m^2-9}{m^2+5m+6} \div \frac{3-m}{m+2}\][/tex]
2. Rewrite the Division as Multiplication:
When dividing by a fraction, you multiply by its reciprocal.
[tex]\[\frac{m^2-9}{m^2+5m+6} \div \frac{3-m}{m+2} = \frac{m^2-9}{m^2+5m+6} \cdot \frac{m+2}{3-m}\][/tex]
3. Factorize the Numerator and Denominator:
Let's factorize each part:
- Numerator [tex]\(m^2 - 9\)[/tex] is a difference of squares:
[tex]\[m^2 - 9 = (m - 3)(m + 3)\][/tex]
- Denominator [tex]\(m^2 + 5m + 6\)[/tex] can be factored as:
[tex]\(m^2 + 5m + 6 = (m + 2)(m + 3)\)[/tex]
4. Rewrite the Expression with Factored Forms:
Substitute the factored forms into the expression:
[tex]\[\frac{(m - 3)(m + 3)}{(m + 2)(m + 3)} \cdot \frac{m + 2}{3 - m}\][/tex]
5. Simplify:
Notice in the expression [tex]\(3 - m\)[/tex], we can rewrite it as [tex]\(-(m - 3)\)[/tex]:
[tex]\[\frac{(m - 3)(m + 3)}{(m + 2)(m + 3)} \cdot \frac{m + 2}{-(m - 3)}\][/tex]
6. Cancellation:
Before multiplying, look for and cancel common factors in the numerator and denominator:
The [tex]\((m + 3)\)[/tex] and [tex]\((m + 2)\)[/tex] terms will cancel each other out in the numerator and denominator:
[tex]\[\frac{(m - 3) \cancel{(m + 3)}}{\cancel{(m + 2)} \cancel{(m + 3)}} \cdot \frac{\cancel{(m + 2)}}{-(m - 3)}\][/tex]
This simplifies to:
[tex]\[\frac{m - 3}{- (m - 3)}\][/tex]
7. Further Simplification:
Recognize that:
[tex]\(\frac{m - 3}{- (m - 3)} = -1\)[/tex]
8. Square Term Simplification:
When squaring [tex]\( \frac{(m-3)}{(m + 2)} \)[/tex]:
[tex]\((m - 3)^2\ / (m + 2)^2\)[/tex]
So the final step would include multiplying by [tex]\(-1\)[/tex]:
[tex]\[\boxed{\frac{(m - 3)^2}{(m + 2)^2} \cdot -1}\][/tex]
Thus, the simplified final expression is:
[tex]\[ \boxed{ -\frac{(m - 3)^2}{(m + 2)^2}} \][/tex]
