`387 xx 387 + 114 xx 114 + 2 xx 387 xx 114 = ?`

Options

A. 250001
B. 251001
C. 260101
D. 261001

Answer

To solve the expression \( 387 \times 387 + 114 \times 114 + 2 \times 387 \times 114 \), we can recognize that it fits the formula for the square of a sum, which is:

\[ (A + B)^2 = A^2 + B^2 + 2AB \]

In our case, we can let:
- \( A = 387 \)
- \( B = 114 \)

Now we can rewrite the expression using the formula:

\[ 387 \times 387 + 114 \times 114 + 2 \times 387 \times 114 = (387 + 114)^2 \]

Next, we calculate \( 387 + 114 \):

\[ 387 + 114 = 501 \]

Now we can substitute this back into our equation:

\[ (387 + 114)^2 = 501^2 \]

Next, we need to calculate \( 501^2 \). We can use the formula for squaring a number:

\[ (a + b)^2 = a^2 + 2ab + b^2 \]

where \( a = 500 \) and \( b = 1 \):

\[ 501^2 = (500 + 1)^2 = 500^2 + 2 \times 500 \times 1 + 1^2 \]

Calculating each part:

1. \( 500^2 = 250000 \)
2. \( 2 \times 500 \times 1 = 1000 \)
3. \( 1^2 = 1 \)

Now, add these together:

\[ 501^2 = 250000 + 1000 + 1 = 251001 \]

Thus, the final answer is:

\[ \boxed{251001} \]