Which expression is equivalent to `\left(\frac{1}{2}a+5\right)\left(\frac{3}{2}a+1\right)` ?
To find the expression that is equivalent to `\left(\frac{1}{2}a+5\right)\left(\frac{3}{2}a+1\right)`, we need to multiply the terms inside the parentheses using the distributive property.
First, we multiply the terms in the first parentheses by the terms in the second parentheses:
`\left(\frac{1}{2}a\right)\left(\frac{3}{2}a\right) + \left(\frac{1}{2}a\right)(1) + (5)\left(\frac{3}{2}a\right) + (5)(1)`
Solving for each term:
`\frac{1}{2}a \cdot \frac{3}{2}a = \frac{3}{4}a^2`
`\frac{1}{2}a \cdot 1 = \frac{1}{2}a`
`5 \cdot \frac{3}{2}a = \frac{15}{2}a`
`5 \cdot 1 = 5`
Now we add the terms together:
`\frac{3}{4}a^2 + \frac{1}{2}a + \frac{15}{2}a + 5`
Combine like terms:
`\frac{3}{4}a^2 + \frac{17}{2}a + 5`
Therefore, the expression that is equivalent to `\left(\frac{1}{2}a+5\right)\left(\frac{3}{2}a+1\right)` is `\frac{3}{4}a^2 + \frac{17}{2}a + 5`.