Simplify:
`(y^3-2y^2+3y-4) (y-1) - (24-3)(y^2-y+1)`

Options

A. `y^4-3y^3-16y^2+14y-17`
B. `y^4-3y^3-16y^2-14y-17`
C. `y^4-3y^3+16y^2+14y-17`
D. `y^4+3y^3-16y^2+14y-17`

Answer

To simplify the expression \((y^3 - 2y^2 + 3y - 4)(y - 1) - (24 - 3)(y^2 - y + 1)\), we will follow these steps:

### Step 1: Expand the first part of the expression
We need to multiply \((y^3 - 2y^2 + 3y - 4)\) by \((y - 1)\).

\[ (y^3 - 2y^2 + 3y - 4)(y - 1) = y^3 \cdot y - y^3 - 2y^2 \cdot y + 2y^2 + 3y \cdot y - 3y - 4 \cdot y + 4 \]

Calculating each term:
- \(y^3 \cdot y = y^4\)
- \(-y^3 \cdot 1 = -y^3\)
- \(-2y^2 \cdot y = -2y^3\)
- \(2y^2 \cdot 1 = 2y^2\)
- \(3y \cdot y = 3y^2\)
- \(-3y \cdot 1 = -3y\)
- \(-4 \cdot y = -4y\)
- \(4 \cdot 1 = 4\)

Combining these gives:
\[ y^4 - 3y^3 + 5y^2 - 7y + 4 \]

### Step 2: Simplify the second part of the expression
Now we simplify \((24 - 3)(y^2 - y + 1)\).

Calculating \(24 - 3\):
\[ 24 - 3 = 21 \]

Now multiply \(21\) by \((y^2 - y + 1)\):
\[ 21(y^2 - y + 1) = 21y^2 - 21y + 21 \]

### Step 3: Combine the results
Now we combine the results from Step 1 and Step 2:
\[ (y^4 - 3y^3 + 5y^2 - 7y + 4) - (21y^2 - 21y + 21) \]

Distributing the negative sign:
\[ y^4 - 3y^3 + 5y^2 - 7y + 4 - 21y^2 + 21y - 21 \]

### Step 4: Combine like terms
Now we combine the like terms:
- For \(y^4\): \(y^4\)
- For \(y^3\): \(-3y^3\)
- For \(y^2\): \(5y^2 - 21y^2 = -16y^2\)
- For \(y\): \(-7y + 21y = 14y\)
- For the constant: \(4 - 21 = -17\)

Thus, the final expression is:
\[ y^4 - 3y^3 - 16y^2 + 14y - 17 \]

### Final Answer
\[ \boxed{y^4 - 3y^3 - 16y^2 + 14y - 17} \]