The remainder obtained when `80x^(3) + 55x^(2) + 20x + 172` is divided by `x +2` is _____

Answer

To find the remainder when the polynomial \(80x^3 + 55x^2 + 20x + 172\) is divided by \(x + 2\), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of a polynomial \(f(x)\) when divided by \(x - c\) is equal to \(f(c)\).

In this case, we need to evaluate the polynomial at \(x = -2\) (since we are dividing by \(x + 2\)).

### Step-by-step Solution:

1. **Identify the polynomial**:
\[ f(x) = 80x^3 + 55x^2 + 20x + 172 \]

2. **Substitute \(x = -2\)** into the polynomial:
\[ f(-2) = 80(-2)^3 + 55(-2)^2 + 20(-2) + 172 \]

3. **Calculate each term**:
- Calculate \(80(-2)^3\):
\[ 80 \times (-8) = -640 \]
- Calculate \(55(-2)^2\):
\[ 55 \times 4 = 220 \]
- Calculate \(20(-2)\):
\[ 20 \times (-2) = -40 \]
- The constant term is \(172\).

4. **Combine all the terms**:
\[ f(-2) = -640 + 220 - 40 + 172 \]

5. **Perform the addition/subtraction**:
- First, combine \(-640 + 220\):
\[ -640 + 220 = -420 \]
- Then, combine \(-420 - 40\):
\[ -420 - 40 = -460 \]
- Finally, combine \(-460 + 172\):
\[ -460 + 172 = -288 \]

6. **Conclusion**:
The remainder when \(80x^3 + 55x^2 + 20x + 172\) is divided by \(x + 2\) is \(-288\).

### Final Answer:
The remainder is \(-288\).