To multiply the two fractions \(-\frac{2}{5}\) and \(\frac{3}{10}\), we follow the rule of multiplying signed numbers and the properties of operations.
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Multiply the numerators: \[ -2 \times 3 = -6 \]
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Multiply the denominators: \[ 5 \times 10 = 50 \]
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Combine the results: \[ -\frac{6}{50} \]
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Simplify the fraction: To simplify \(-\frac{6}{50}\), we can find the greatest common divisor (GCD) of 6 and 50, which is 2.
Divide both the numerator and the denominator by their GCD: \[ \frac{-6 \div 2}{50 \div 2} = \frac{-3}{25} \]
Thus, the product of \(-\frac{2}{5}\) and \(\frac{3}{10}\) in simplest form is: \[ \boxed{-\frac{3}{25}} \]
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