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:
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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 \( (x + 3)(x - 2) \)
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Rewrite the original expression using these factors: \[ \frac{5x(x + 2)}{(x + 3)(x + 4)} \div \frac{(x - 2)(x + 2)}{(x + 3)(x - 2)} \]
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Change the division to multiplication by the reciprocal: \[ \frac{5x(x + 2)}{(x + 3)(x + 4)} \times \frac{(x + 3)(x - 2)}{(x - 2)(x + 2)} \]
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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.