To multiply \( 5x \) by the expression \( (3x^2 + 2x - 7) \), we will distribute \( 5x \) to each term inside the parentheses.
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Multiply \( 5x \) by \( 3x^2 \): \[ 5x \cdot 3x^2 = 15x^{3} \]
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Multiply \( 5x \) by \( 2x \): \[ 5x \cdot 2x = 10x^{2} \]
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Multiply \( 5x \) by \( -7 \): \[ 5x \cdot (-7) = -35x \]
Now, combine all the results: \[ 15x^3 + 10x^2 - 35x \]
So, the final result is: \[ \boxed{15x^3 + 10x^2 - 35x} \]
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