Subtract :
`5y^(4) - 3y^(3) + 2y^(2) + y - 1` from `4y^(4) - 2y^(3) - 6y^(2) - y + 5`

Answer

To solve the problem of subtracting the algebraic expression \( 5y^{4} - 3y^{3} + 2y^{2} + y - 1 \) from \( 4y^{4} - 2y^{3} - 6y^{2} - y + 5 \), we will follow these steps:

### Step 1: Write the expression for subtraction
We start by writing the expression for the subtraction:
\[ (4y^{4} - 2y^{3} - 6y^{2} - y + 5) - (5y^{4} - 3y^{3} + 2y^{2} + y - 1) \]

### Step 2: Distribute the negative sign
Next, we distribute the negative sign across the second expression:
\[ 4y^{4} - 2y^{3} - 6y^{2} - y + 5 - 5y^{4} + 3y^{3} - 2y^{2} - y + 1 \]
This changes the signs of the terms in the second expression.

### Step 3: Combine like terms
Now we will combine the like terms:
- For \( y^{4} \) terms: \( 4y^{4} - 5y^{4} = -1y^{4} \)
- For \( y^{3} \) terms: \( -2y^{3} + 3y^{3} = 1y^{3} \)
- For \( y^{2} \) terms: \( -6y^{2} - 2y^{2} = -8y^{2} \)
- For \( y \) terms: \( -y - y = -2y \)
- For the constant terms: \( 5 + 1 = 6 \)

### Step 4: Write the final expression
Putting it all together, we have:
\[ -y^{4} + y^{3} - 8y^{2} - 2y + 6 \]

### Final Answer:
Thus, the result of the subtraction is:
\[ -y^{4} + y^{3} - 8y^{2} - 2y + 6 \]

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