What is the value of
`a(a+b^2+c)+b^2(a^2+b^2+c^2)-c(a+b^2)`, when a = 1, b = -3 and c = -2?

Options

A. 138
B. 154
C. 162
D. 176

Answer

To find the value of the expression \( a(a+b^2+c) + b^2(a^2+b^2+c^2) - c(a+b^2) \) when \( a = 1 \), \( b = -3 \), and \( c = -2 \), we will substitute the values of \( a \), \( b \), and \( c \) step by step.

### Step 1: Substitute the values into the expression
We start with the expression:
\[ a(a+b^2+c) + b^2(a^2+b^2+c^2) - c(a+b^2) \]
Substituting \( a = 1 \), \( b = -3 \), and \( c = -2 \):
\[ 1(1 + (-3)^2 + (-2)) + (-3)^2(1^2 + (-3)^2 + (-2)^2) - (-2)(1 + (-3)^2) \]

### Step 2: Calculate \( b^2 \) and \( c^2 \)
Calculate \( b^2 \) and \( c^2 \):
\[ b^2 = (-3)^2 = 9 \]
\[ c^2 = (-2)^2 = 4 \]

### Step 3: Substitute \( b^2 \) and \( c^2 \) into the expression
Now substitute \( b^2 \) and \( c^2 \) back into the expression:
\[ 1(1 + 9 - 2) + 9(1 + 9 + 4) - (-2)(1 + 9) \]

### Step 4: Simplify the first term
Calculate the first term:
\[ 1(1 + 9 - 2) = 1(8) = 8 \]

### Step 5: Simplify the second term
Calculate the second term:
\[ 9(1 + 9 + 4) = 9(14) = 126 \]

### Step 6: Simplify the third term
Calculate the third term:
\[ -(-2)(1 + 9) = 2(10) = 20 \]

### Step 7: Combine all the terms
Now combine all the terms:
\[ 8 + 126 + 20 = 154 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{154} \]