Expand and Simplify: (3/4+2a)(8a+9ba)-(5+a)(3/2a+b)

To solve this problem, we'll simplify the expression step by step using the distributive law.

Step 1: Apply the distributive property to the first part of the expression: $(\frac{3}{4} + 2a)(8a + 9ba)$.

• Distribute $\frac{3}{4}$ to $8a$ and $9ba$: $\frac{3}{4} \times 8a = 6a$ and $\frac{3}{4} \times 9ba = \frac{27}{4}ab$.
• Distribute $2a$ to $8a$ and $9ba$: $2a \times 8a = 16a^2$ and $2a \times 9ba = 18a^2b$.

The first part expands to: $6a + \frac{27}{4}ab + 16a^2 + 18a^2b$.

Step 2: Apply the distributive property to the second part of the expression: $(5 + a)(\frac{3}{2}a + b)$.

• Distribute $5$ to $\frac{3}{2}a$ and $b$: $5 \times \frac{3}{2}a = \frac{15}{2}a$ and $5 \times b = 5b$.
• Distribute $a$ to $\frac{3}{2}a$ and $b$: $a \times \frac{3}{2}a = \frac{3}{2}a^2$ and $a \times b = ab$.

The second part expands to: $\frac{15}{2}a + 5b + \frac{3}{2}a^2 + ab$.

Step 3: Simplify the expression by subtracting the second part from the first:

• Combine like terms: $6a - \frac{15}{2}a$ and $\frac{27}{4}ab - ab$.
• Subtract constants and like terms: - $6a - \frac{15}{2}a = -\frac{3}{2}a$. - $\frac{27}{4}ab - ab = \frac{27}{4}ab - \frac{4}{4}ab = \frac{23}{4}ab$. - $(16a^2 - \frac{3}{2}a^2) + (18a^2b - 5b)$.

The full simplified expression is: $-\frac{3}{2}a + \frac{23}{4}ab + \left(16a^2 - \frac{3}{2}a^2\right) + (18a^2b - 5b)$.

Recognize that $16a^2 - \frac{3}{2}a^2 = \frac{32}{2}a^2 - \frac{3}{2}a^2 = \frac{29}{2}a^2$, the final answer is:

The simplified expression is: $-\frac{3}{2}a + 5\frac{3}{4}ab + 14\frac{1}{2}a^2 + (18a^2-5)b$.