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 ?

Q: 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 ?

To solve the expression $ a(a + b^2 + c) + b^2(a^2 + b^2 + c^2) - c(a + b^2) $ with the given values $ a = 1 $, $ b = -3 $, and $ c = -2 $, we will substitute these values into the expression and simplify step by step. 1. **Substitute the values into the expression:** \[ 1(1 + (-3)^2 + (-2)) + (-3)^2(1^2 + (-3)^2 + (-2)^2) - (-2)(1 + (-3)^2) \] 2. **Calculate each part of the expression:** - Calculate $ b^2 $ and substitute: \[ (-3)^2 = 9 \] - Substitute into the expression: \[ 1(1 + 9 - 2) + 9(1 + 9 + 4) + 2(1 + 9) \] 3. **Simplify each term:** - First term: \[ 1(1 + 9 - 2) = 1 \times 8 = 8 \] - Second term: \[ 9(1 + 9 + 4) = 9 \times 14 = 126 \] - Third term: \[ 2(1 + 9) = 2 \times 10 = 20 \] 4. **Combine all terms:** \[ 8 + 126 + 20 = 154 \] Therefore, the value of the expression is 154, which corresponds to Option 2. **Explanation of why other options are incorrect:** - **Option 1 (138):** This value is incorrect because the calculation of each term and their sum does not result in 138. The correct calculation, as shown, results in 154. - **Option 3 (162):** This value is incorrect because the sum of the calculated terms does not reach 162. The correct sum is 154. - **Option 4 (176):** This value is incorrect because the total of the evaluated expression does not equal 176. The correct total is 154. In conclusion, the correct answer is Option 2, which is 154, as demonstrated by the detailed calculation and logical reasoning provided.

Question

Accepted Answer

To solve the expression $ a(a + b^2 + c) + b^2(a^2 + b^2 + c^2) - c(a + b^2) $ with the given values $ a = 1 $, $ b = -3 $, and $ c = -2 $, we will substitute these values into the expression and simplify step by step.

1. **Substitute the values into the expression:**

$$
   1(1 + (-3)^2 + (-2)) + (-3)^2(1^2 + (-3)^2 + (-2)^2) - (-2)(1 + (-3)^2)
   $$

2. **Calculate each part of the expression:**

- Calculate $ b^2 $ and substitute:
     $$
     (-3)^2 = 9
     $$

- Substitute into the expression:
     $$
     1(1 + 9 - 2) + 9(1 + 9 + 4) + 2(1 + 9)
     $$

3. **Simplify each term:**

- First term:
     $$
     1(1 + 9 - 2) = 1 \times 8 = 8
     $$

- Second term:
     $$
     9(1 + 9 + 4) = 9 \times 14 = 126
     $$

- Third term:
     $$
     2(1 + 9) = 2 \times 10 = 20
     $$

4. **Combine all terms:**

$$
   8 + 126 + 20 = 154
   $$

Therefore, the value of the expression is 154, which corresponds to Option 2.

**Explanation of why other options are incorrect:**

- **Option 1 (138):** This value is incorrect because the calculation of each term and their sum does not result in 138. The correct calculation, as shown, results in 154.

- **Option 3 (162):** This value is incorrect because the sum of the calculated terms does not reach 162. The correct sum is 154.

- **Option 4 (176):** This value is incorrect because the total of the evaluated expression does not equal 176. The correct total is 154.

In conclusion, the correct answer is Option 2, which is 154, as demonstrated by the detailed calculation and logical reasoning provided.