The expression `3(a^2+1)^2+(2(a-1)(a^2+1)-5(a-1)^2-4(0.75a^4+3a-1)` when sinplified redues to

Options

A. `2a^(3)-a^(2)`
B. `2a^(2)-a^(3)`
C. `2a^(3)`
D. `2a^(2)`

Answer

To simplify the expression \( 3(a^2 + 1)^2 + 2(a - 1)(a^2 + 1) - 5(a - 1)^2 - 4(0.75a^4 + 3a - 1) \), we will follow these steps:

### Step 1: Expand \( 3(a^2 + 1)^2 \)
Using the formula \( (x + y)^2 = x^2 + 2xy + y^2 \):
\[ 3(a^2 + 1)^2 = 3(a^4 + 2a^2 + 1) = 3a^4 + 6a^2 + 3 \]

### Step 2: Expand \( 2(a - 1)(a^2 + 1) \)
Distributing the terms:
\[ 2(a - 1)(a^2 + 1) = 2(a^3 + a - a^2 - 1) = 2a^3 + 2a - 2a^2 - 2 \]

### Step 3: Expand \( -5(a - 1)^2 \)
Using the formula \( (x - y)^2 = x^2 - 2xy + y^2 \):
\[ -5(a - 1)^2 = -5(a^2 - 2a + 1) = -5a^2 + 10a - 5 \]

### Step 4: Expand \( -4(0.75a^4 + 3a - 1) \)
Distributing the \( -4 \):
\[ -4(0.75a^4 + 3a - 1) = -3a^4 - 12a + 4 \]

### Step 5: Combine all the expanded terms
Now we combine all the terms from Steps 1 to 4:
\[ 3a^4 + 6a^2 + 3 + 2a^3 + 2a - 2a^2 - 2 - 5a^2 + 10a - 5 - 3a^4 - 12a + 4 \]

### Step 6: Group like terms
Now we will group the terms by their degrees:
- \( a^4 \) terms: \( 3a^4 - 3a^4 = 0 \)
- \( a^3 \) terms: \( 2a^3 \)
- \( a^2 \) terms: \( 6a^2 - 2a^2 - 5a^2 = -a^2 \)
- \( a \) terms: \( 2a + 10a - 12a = 0 \)
- Constant terms: \( 3 - 2 - 5 + 4 = 0 \)

### Final Result
Putting it all together, we have:
\[ 2a^3 - a^2 + 0 + 0 = 2a^3 - a^2 \]

Thus, the simplified expression is:
\[ \boxed{2a^3 - a^2} \]