We are tasked with simplifying the given rational expression:
\[
\frac{x^2 + 3x - 10}{3x^2 + 13x - 10} \div \frac{x^2 - 5x + 6}{2x^2 + 6x - 36}
\]
Step 1: Rewrite the division as multiplication by the reciprocal
The division of two fractions can be rewritten as multiplication by the reciprocal of the second fraction:
\[
\frac{x^2 + 3x - 10}{3x^2 + 13x - 10} \div \frac{x^2 - 5x + 6}{2x^2 + 6x - 36} = \frac{x^2 + 3x - 10}{3x^2 + 13x - 10} \cdot \frac{2x^2 + 6x - 36}{x^2 - 5x + 6}
\]
Step 2: Factor each polynomial in the expression
We will factor each polynomial in the numerator and denominator.
#### Factor \(x^2 + 3x - 10\):
This is a quadratic polynomial. We look for two numbers that multiply to \(-10\) and add to \(3\). These numbers are \(5\) and \(-2\):
\[
x^2 + 3x - 10 = (x + 5)(x - 2)
\]
#### Factor \(3x^2 + 13x - 10\):
This is also a quadratic polynomial. We use the AC method or trial and error. We need two numbers that multiply to \(3 \cdot (-10) = -30\) and add to \(13\). These numbers are \(15\) and \(-2\):
\[
3x^2 + 13x - 10 = (3x - 2)(x + 5)
\]
#### Factor \(x^2 - 5x + 6\):
This is a quadratic polynomial. We look for two numbers that multiply to \(6\) and add to \(-5\). These numbers are \(-3\) and \(-2\):
\[
x^2 - 5x + 6 = (x - 3)(x - 2)
\]
#### Factor \(2x^2 + 6x - 36\):
First, factor out the greatest common factor (GCF), which is \(2\):
\[
2x^2 + 6x - 36 = 2(x^2 + 3x - 18)
\]
Now, factor the quadratic \(x^2 + 3x - 18\). We need two numbers that multiply to \(-18\) and add to \(3\). These numbers are \(6\) and \(-3\):
\[
x^2 + 3x - 18 = (x + 6)(x - 3)
\]
Thus:
\[
2x^2 + 6x - 36 = 2(x + 6)(x - 3)
\]
Step 3: Substitute the factored forms back into the expression
Substitute the factored forms into the original expression:
\[
\frac{x^2 + 3x - 10}{3x^2 + 13x - 10} \cdot \frac{2x^2 + 6x - 36}{x^2 - 5x + 6} = \frac{(x + 5)(x - 2)}{(3x - 2)(x + 5)} \cdot \frac{2(x + 6)(x - 3)}{(x - 3)(x - 2)}
\]
Step 4: Simplify the expression
Cancel out common factors in the numerator and denominator:
1. Cancel \((x + 5)\) in the numerator and denominator:
\[
\frac{(x + 5)(x - 2)}{(3x - 2)(x + 5)} \cdot \frac{2(x + 6)(x - 3)}{(x - 3)(x - 2)} = \frac{x - 2}{3x - 2} \cdot \frac{2(x + 6)(x - 3)}{(x - 3)(x - 2)}
\]
2. Cancel \((x - 2)\) in the numerator and denominator:
\[
\frac{x - 2}{3x - 2} \cdot \frac{2(x + 6)(x - 3)}{(x - 3)(x - 2)} = \frac{1}{3x - 2} \cdot \frac{2(x + 6)(x - 3)}{x - 3}
\]
3. Cancel \((x - 3)\) in the numerator and denominator:
\[
\frac{1}{3x - 2} \cdot \frac{2(x + 6)(x - 3)}{x - 3} = \frac{1}{3x - 2} \cdot 2(x + 6)
\]
Step 5: Write the final simplified expression
Combine the remaining terms:
\[
\frac{1}{3x - 2} \cdot 2(x + 6) = \frac{2(x + 6)}{3x - 2}
\]
Final Answer:
\[
\boxed{\frac{2(x + 6)}{3x - 2}}
\]
Parent Tip: Review the logic above to help your child master the concept of rational expressions.
