If `a^(3) - b^(3) = 210` and `a-b = 5`, then `( a+ b)^(2) - ab` is equal to `:`

Options

A. 52
B. 42
C. 32
D. 38

Answer

To solve the problem, we start with the given equations:

1. \( a^3 - b^3 = 210 \)
2. \( a - b = 5 \)

We need to find the value of \( (a + b)^2 - ab \).

### Step 1: Use the identity for the difference of cubes
The difference of cubes can be factored using the identity:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Substituting the known values into this identity:
\[ 210 = (5)(a^2 + ab + b^2) \]

### Step 2: Solve for \( a^2 + ab + b^2 \)
Now we can solve for \( a^2 + ab + b^2 \):
\[ a^2 + ab + b^2 = \frac{210}{5} = 42 \]

### Step 3: Use the identity for \( (a + b)^2 \)
We know that:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
We can express \( a^2 + b^2 \) in terms of \( a^2 + ab + b^2 \):
\[ a^2 + b^2 = (a^2 + ab + b^2) - ab = 42 - ab \]

### Step 4: Substitute into \( (a + b)^2 - ab \)
Now we can substitute \( a^2 + b^2 \) into the equation:
\[ (a + b)^2 - ab = (a^2 + b^2 + 2ab) - ab = a^2 + b^2 + ab \]
Substituting \( a^2 + b^2 = 42 - ab \):
\[ (a + b)^2 - ab = (42 - ab) + ab = 42 \]

### Final Answer
Thus, the value of \( (a + b)^2 - ab \) is:
\[ \boxed{42} \]