To multiply the expression \((3a + 1)(5a - 7)\), we can use the distributive property (also known as the FOIL method for binomials).
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First: Multiply the first terms: \[ 3a \cdot 5a = 15a^2 \]
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Outer: Multiply the outer terms: \[ 3a \cdot (-7) = -21a \]
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Inner: Multiply the inner terms: \[ 1 \cdot 5a = 5a \]
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Last: Multiply the last terms: \[ 1 \cdot (-7) = -7 \]
Now, combine all these results: \[ 15a^2 - 21a + 5a - 7 \]
Next, combine the like terms (-21a and 5a): \[ 15a^2 - 16a - 7 \]
Thus, the simplified expression is: \[ \boxed{15a^2 - 16a - 7} \]