To find an expression equivalent to \((\frac{1}{2}a + 5)(\frac{3}{2}a + 1)\), we will use the distributive property (also known as the FOIL method for binomials) to expand the expression.
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First: Multiply the first terms: \[ \frac{1}{2}a \cdot \frac{3}{2}a = \frac{3}{4}a^2 \]
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Outer: Multiply the outer terms: \[ \frac{1}{2}a \cdot 1 = \frac{1}{2}a \]
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Inner: Multiply the inner terms: \[ 5 \cdot \frac{3}{2}a = \frac{15}{2}a \]
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Last: Multiply the last terms: \[ 5 \cdot 1 = 5 \]
Now, combine all these results: \[ \frac{3}{4}a^2 + \frac{1}{2}a + \frac{15}{2}a + 5 \]
Next, combine the \(a\) terms: \[ \frac{1}{2}a + \frac{15}{2}a = \frac{16}{2}a = 8a \]
Thus, the combined expression is: \[ \frac{3}{4}a^2 + 8a + 5 \]
Therefore, the expression equivalent to \((\frac{1}{2}a + 5)(\frac{3}{2}a + 1)\) is: \[ \frac{3}{4}a^2 + 8a + 5 \]