Certainly! Let's go through the problem step-by-step:
Given the expression:
[tex]\[
\frac{5}{7m^2 + m + 3} - \frac{1}{7m^2 - 18m + 8}
\][/tex]
### Step 1: Factor the denominators
First, we need to factor the denominators.
For [tex]\(7m^2 + m + 3\)[/tex]:
This polynomial does not factor into simpler rational expressions with integer coefficients, it remains as is:
[tex]\[
7m^2 + m + 3
\][/tex]
For [tex]\(7m^2 - 18m + 8\)[/tex]:
This polynomial can be factored. It factors into:
[tex]\[
7m^2 - 18m + 8 = (m - 2)(7m - 4)
\][/tex]
So now, we rewrite the expression with the factored denominator:
[tex]\[
\frac{5}{7m^2 + m + 3} - \frac{1}{(m - 2)(7m - 4)}
\][/tex]
### Step 2: Find a common denominator
The common denominator for the fractions will be the product of the two distinct denominators:
[tex]\[
(7m^2 + m + 3)((m - 2)(7m - 4))
\][/tex]
### Step 3: Rewrite the fractions with the common denominator
To express the fractions with this common denominator, we adjust the numerators accordingly:
For the first term, [tex]\( \frac{5}{7m^2 + m + 3} \)[/tex]:
[tex]\[
\frac{5 \cdot (m - 2)(7m - 4)}{(7m^2 + m + 3)((m - 2)(7m - 4))}
\][/tex]
For the second term, [tex]\( \frac{1}{(m - 2)(7m - 4)} \)[/tex]:
[tex]\[
\frac{1 \cdot (7m^2 + m + 3)}{(7m^2 + m + 3)((m - 2)(7m - 4))}
\][/tex]
Combining these, we get:
[tex]\[
\frac{5(m - 2)(7m - 4) - (7m^2 + m + 3)}{(7m^2 + m + 3)((m - 2)(7m - 4))}
\][/tex]
### Step 4: Simplify the numerator
Now we expand and simplify the numerator:
[tex]\[
5(m - 2)(7m - 4) = 5(7m^2 - 4m - 14m + 8) = 5(7m^2 - 18m + 8) = 35m^2 - 90m + 40
\][/tex]
Therefore,
[tex]\[
5(m - 2)(7m - 4) = 35m^2 - 90m + 40
\][/tex]
The numerator becomes:
[tex]\[
35m^2 - 90m + 40 - (7m^2 + m + 3)
\][/tex]
Simplify the expression by combining like terms:
[tex]\[
35m^2 - 90m + 40 - 7m^2 - m - 3 = 28m^2 - 91m + 37
\][/tex]
So, the simplified numerator is:
[tex]\[
28m^2 - 91m + 37
\][/tex]
### Step 5: Write the final simplified fraction
Putting it all together, the simplified fraction is:
[tex]\[
\frac{28m^2 - 91m + 37}{(7m^2 + m + 3)((m - 2)(7m - 4))}
\][/tex]
### Final Answer
[tex]\[
\boxed{\frac{28m^2 - 91m + 37}{(7m^2 + m + 3)((m - 2)(7m - 4))}}
\][/tex]
