Express in simplest form: 5x2+10x/x2+7x+12÷x2−4/x2+x−6

Question

Express in simplest form:  5x2+10x/x2+7x+12÷x2−4/x2+x−6
 .(1 point)
Responses

54
Start Fraction 5 over 4 End Fraction

5x+4
Start Fraction 5 over x plus 4 End Fraction

5xx+4
Start Fraction 5 x over x plus 4 End Fraction

x+45x

GPT-4o mini · Answer

To simplify the expression $ \frac{5x^2 + 10x}{x^2 + 7x + 12} \div \frac{x^2 - 4}{x^2 + x - 6} $, we can follow these steps: 1. **Factor the polynomials where possible**. - $ 5x^2 + 10x = 5x(x + 2) $ - $ x^2 + 7x + 12 $ factors to $ (x + 3)(x + 4) $ - $ x^2 - 4 $ is a difference of squares, so it factors to $ (x - 2)(x + 2) $ - $ x^2 + x - 6 $ factors to $ (x + 3)(x - 2) $ 2. **Rewrite the original expression using these factors**: $$ \frac{5x(x + 2)}{(x + 3)(x + 4)} \div \frac{(x - 2)(x + 2)}{(x + 3)(x - 2)} $$ 3. **Change the division to multiplication by the reciprocal**: $$ \frac{5x(x + 2)}{(x + 3)(x + 4)} \times \frac{(x + 3)(x - 2)}{(x - 2)(x + 2)} $$ 4. **Simplify the expression**. Cancel out common factors from the numerator and denominator: - The $ (x + 2) $ cancels. - The $ (x + 3) $ cancels. Now we have: $$ \frac{5x \cdot (x - 2)}{(x + 4) \cdot (x - 2)} $$ Again, the $ (x - 2) $ cancels: $$ \frac{5x}{x + 4} $$ So, the simplest form of the original expression is: $$ \frac{5x}{x + 4} $$ This matches the choice: **Start Fraction 5 x over x plus 4 End Fraction**, which is the correct answer.