The given problem asks you to multiply two complex numbers and express the result in terms of $i$. The expression is:

$(10 + 7i) \cdot (10 + 7i)$

This is equivalent to squaring the complex number $10 + 7i$:

$(10 + 7i)^2$

To expand this, we use the distributive property (FOIL):

$(10 + 7i)(10 + 7i) = 10^2 + 2(10)(7i) + (7i)^2$

Now, calculating each term:

• $10^2 = 100$
• $2(10)(7i) = 140i$
• $(7i)^2 = 49i^2$

Recall that $i^2 = -1$, so:

$49i^2 = 49(-1) = -49$

Now, combine all the terms:

$100 + 140i - 49 = (100 - 49) + 140i = 51 + 140i$

Thus, the final simplified result is:

$51 + 140i$

Would you like further explanation on this? Here are a few questions related to complex number multiplication that might help you:

1. What is the general formula for multiplying two complex numbers?
2. How does the imaginary unit $i$ behave under multiplication?
3. How does $i^2 = -1$ affect complex number operations?
4. Can complex numbers be visualized geometrically in the complex plane?
5. How do powers of $i$ cycle through different values?

Tip: When multiplying complex numbers, always apply the distributive property and remember that $i^2 = -1$.