Certainly! Let's go through the steps to subtract the given rational expressions and simplify the result.
We start with the following expressions:
[tex]\[
\frac{3m - 18}{m^2 - 10m + 24} - \frac{m + 8}{4 - m}
\][/tex]
Step 1: Simplify the second fraction, if possible
Notice that the denominator [tex]\(4 - m\)[/tex] can be written as [tex]\( -(m - 4)\)[/tex]. Thus, the second fraction becomes:
[tex]\[
\frac{m + 8}{4 - m} = \frac{m + 8}{-(m - 4)} = -\frac{m + 8}{m - 4}
\][/tex]
Now, our expression looks like this:
[tex]\[
\frac{3m - 18}{m^2 - 10m + 24} - \left( -\frac{m + 8}{m - 4} \right)
\][/tex]
Which simplifies to:
[tex]\[
\frac{3m - 18}{m^2 - 10m + 24} + \frac{m + 8}{m - 4}
\][/tex]
Step 2: Factor the denominators, if necessary (factoring the quadratic expression)
The denominator [tex]\(m^2 - 10m + 24\)[/tex] can be factored by looking for factors of 24 that sum up to -10. The correct factors are -6 and -4:
[tex]\[
m^2 - 10m + 24 = (m - 6)(m - 4)
\][/tex]
So, our expression is now:
[tex]\[
\frac{3m - 18}{(m - 6)(m - 4)} + \frac{m + 8}{m - 4}
\][/tex]
Step 3: Combine the fractions by finding a common denominator
The common denominator in this case is [tex]\((m - 6)(m - 4)\)[/tex]. We need to express [tex]\(\frac{m + 8}{m - 4}\)[/tex] with this common denominator:
[tex]\[
\frac{m + 8}{m - 4} = \frac{(m + 8)(m - 6)}{(m - 6)(m - 4)}
\][/tex]
So, now we have:
[tex]\[
\frac{3m - 18}{(m - 6)(m - 4)} + \frac{(m + 8)(m - 6)}{(m - 6)(m - 4)}
\][/tex]
Step 4: Add the numerators over the common denominator
The numerators are [tex]\(3m - 18\)[/tex] and [tex]\((m + 8)(m - 6)\)[/tex]. First, expand [tex]\((m + 8)(m - 6)\)[/tex]:
[tex]\[
(m + 8)(m - 6) = m^2 - 6m + 8m - 48 = m^2 + 2m - 48
\][/tex]
Now add the numerators:
[tex]\[
\frac{3m - 18 + (m^2 + 2m - 48)}{(m - 6)(m - 4)} = \frac{m^2 + 2m - 48 + 3m - 18}{(m - 6)(m - 4)} = \frac{m^2 + 5m - 66}{(m - 6)(m - 4)}
\][/tex]
Step 5: Simplify the resulting expression if possible
To ensure the simplified expression has no common factors in the numerator and denominator, check if the numerator [tex]\(m^2 + 5m - 66\)[/tex] factors further.
It does not factor any further in a real number field, so the rational expression is completely simplified.
Final Answer:
[tex]\[
\frac{m^2 + 5m - 66}{(m - 6)(m - 4)}
\][/tex]
This can be further verified to simplify to [tex]\(\frac{m + 11}{m - 4}\)[/tex] upon performing polynomial long division.
Hence, the final simplified form is:
[tex]\[
\boxed{\frac{m + 11}{m - 4}}
\][/tex]
