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

Answer

To solve the problem of subtracting \(6a + 3\) from \(a^3 - 3a^2 + 4a + 1\), we will follow these steps:

### Step 1: Write the expression for subtraction
We need to subtract \(6a + 3\) from \(a^3 - 3a^2 + 4a + 1\). This can be written as:
\[ a^3 - 3a^2 + 4a + 1 - (6a + 3) \]

### Step 2: Distribute the negative sign
Next, we distribute the negative sign across the terms in the parentheses:
\[ a^3 - 3a^2 + 4a + 1 - 6a - 3 \]

### Step 3: Combine like terms
Now, we will combine the like terms:
- The \(a^3\) term: \(a^3\)
- The \(a^2\) term: \(-3a^2\)
- The \(a\) terms: \(4a - 6a = -2a\)
- The constant terms: \(1 - 3 = -2\)

Putting it all together, we have:
\[ a^3 - 3a^2 - 2a - 2 \]

### Final Answer
Thus, the result of subtracting \(6a + 3\) from \(a^3 - 3a^2 + 4a + 1\) is:
\[ \boxed{a^3 - 3a^2 - 2a - 2} \]