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.
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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, \(3q^2 + 13q - 10 = (3q - 2)(q + 5)\).
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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.
- Expanding \((5q + 2)(4q - 15)\):
\[ 20q^2 - 75q + 8q - 30 = 20q^2 - 67q - 30 \]
- 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.