Question

In this subtraction​ problem, the rational expression that follows the subtraction sign has a numerator with more than one term. Be careful with signs and find the difference.
StartFraction 5 q plus 2 Over 3 q squared plus 13 q minus 10 EndFraction minus StartFraction 4 q plus 1 Over 4 q squared plus 5 q minus 75 EndFraction

Answer 1

To solve the subtraction problem involving rational expressions, we will first simplify the expressions and then find a common denominator to perform the subtraction.

The problem is:

\[
\frac{5q + 2}{3q^2 + 13q - 10} - \frac{4q + 1}{4q^2 + 5q - 75}
\]

### Step 1: Factor the denominators.

1. **First Denominator: \(3q^2 + 13q - 10\)**

We will factor \(3q^2 + 13q - 10\). We are looking for two numbers that multiply to \(-30\) (the product of \(3\) and \(-10\)) and add to \(13\). The factors we can use are \(15\) and \(-2\):

\[
3q^2 + 15q - 2q - 10 \Rightarrow 3q(q + 5) - 2(q + 5) \Rightarrow (3q - 2)(q + 5)
\]

Thus, \(3q^2 + 13q - 10 = (3q - 2)(q + 5)\).

2. **Second Denominator: \(4q^2 + 5q - 75\)**

We will factor \(4q^2 + 5q - 75\). We are looking for two numbers that multiply to \(-300\) (the product of \(4\) and \(-75\)) and add to \(5\). The factors we can use are \(20\) and \(-15\):

\[
4q^2 + 20q - 15q - 75 \Rightarrow 4q(q + 5) - 15(q + 5) \Rightarrow (4q - 15)(q + 5)
\]

Thus, \(4q^2 + 5q - 75 = (4q - 15)(q + 5)\).

### Step 2: Rewrite the subtraction with factored denominators.

Now we can write the problem as:

\[
\frac{5q + 2}{(3q - 2)(q + 5)} - \frac{4q + 1}{(4q - 15)(q + 5)}
\]

### Step 3: Find a common denominator.

The common denominator will be \((3q - 2)(4q - 15)(q + 5)\).

### Step 4: Rewrite each term with the common denominator.

- For \( \frac{5q + 2}{(3q - 2)(q + 5)}\):

\[
\frac{5q + 2}{(3q - 2)(q + 5)} \cdot \frac{(4q - 15)}{(4q - 15)} = \frac{(5q + 2)(4q - 15)}{(3q - 2)(4q - 15)(q + 5)}
\]

- For \( \frac{4q + 1}{(4q - 15)(q + 5)}\):

\[
\frac{4q + 1}{(4q - 15)(q + 5)} \cdot \frac{(3q - 2)}{(3q - 2)} = \frac{(4q + 1)(3q - 2)}{(3q - 2)(4q - 15)(q + 5)}
\]

### Step 5: Combine the two fractions.

Now we can write

\[
\frac{(5q + 2)(4q - 15) - (4q + 1)(3q - 2)}{(3q - 2)(4q - 15)(q + 5)}
\]

### Step 6: Expand the numerators.

1. **Expanding \((5q + 2)(4q - 15)\)**:

\[
20q^2 - 75q + 8q - 30 = 20q^2 - 67q - 30
\]

2. **Expanding \((4q + 1)(3q - 2)\)**:

\[
12q^2 - 8q + 3q - 2 = 12q^2 - 5q - 2
\]

### Step 7: Combine the expanded numerators and simplify.

Combine the two results in the numerator:

\[
20q^2 - 67q - 30 - (12q^2 - 5q - 2) = 20q^2 - 67q - 30 - 12q^2 + 5q + 2
\]

This simplifies to:

\[
(20q^2 - 12q^2) + (-67q + 5q) + (-30 + 2) = 8q^2 - 62q - 28
\]

### Final Form:

We can then write our final answer as:

\[
\frac{8q^2 - 62q - 28}{(3q - 2)(4q - 15)(q + 5)}
\]

You can also factor \(8q^2 - 62q - 28\) if needed, but at this stage, this expression is valid as the answer.