Simplify the expression:

[tex]\[ \frac{5}{7m^2 - 1 + m + 4} - \frac{1}{7m^2 - 18m + 8} \][/tex]

Completely factor and simplify your answer as much as possible.

Answer: [tex]\(\square\)[/tex]



Answer :

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]