Add `4a^2b-2b^2c+3c^2a+4abc-5 and a^2b+b^2-2c^2+8ab`

Answer

To solve the problem of adding the two algebraic expressions \(4a^2b - 2b^2c + 3c^2a + 4abc - 5\) and \(a^2b + b^2 - 2c^2 + 8ab\), we will follow these steps:

### Step 1: Write down the expressions
We have:
1. First expression: \(4a^2b - 2b^2c + 3c^2a + 4abc - 5\)
2. Second expression: \(a^2b + b^2 - 2c^2 + 8ab\)

### Step 2: Arrange the expressions for addition
We will align the two expressions so that like terms are in the same column:
\[ \begin{array}{r} 4a^2b \\ -2b^2c \\ 3c^2a \\ 4abc \\ -5 \\ \hline \end{array} + \begin{array}{r} a^2b \\ b^2 \\ -2c^2 \\ 8ab \\ \hline \end{array} \]

### Step 3: Identify and combine like terms
Now, we will combine the like terms from both expressions:
- For \(a^2b\): \(4a^2b + a^2b = 5a^2b\)
- For \(b^2c\): \(-2b^2c + b^2 = -2b^2c + 1b^2 = -2b^2c\) (since there is no like term for \(b^2\) in the first expression)
- For \(c^2a\): \(3c^2a\) (no like term in the second expression)
- For \(abc\): \(4abc + 8ab\) (no like term in the first expression)
- For the constant terms: \(-5\) (no like term in the second expression)
- For \(c^2\): \(-2c^2\) (no like term in the first expression)

### Step 4: Write the final expression
Combining all the results, we have:
\[ 5a^2b - 2b^2c + 3c^2a + 4abc + b^2 - 2c^2 - 5 \]

### Final Answer
Thus, the final result of adding the two expressions is:
\[ 5a^2b - 2b^2c + 3c^2a + 4abc + b^2 - 2c^2 - 5 \]