To multiply the complex numbers \((-10 + 5i)\) and \((-10 - 5i)\), we can use the formula for multiplying two binomials, which is similar to the distributive property (also known as the FOIL method):
\[ (a + b)(c + d) = ac + ad + bc + bd \]
For our specific case:
- \(a = -10\)
- \(b = 5i\)
- \(c = -10\)
- \(d = -5i\)
Now, we multiply:
\[ (-10 + 5i)(-10 - 5i) = (-10)(-10) + (-10)(-5i) + (5i)(-10) + (5i)(-5i) \]
Calculating each term:
- \((-10)(-10) = 100\)
- \((-10)(-5i) = 50i\)
- \((5i)(-10) = -50i\)
- \((5i)(-5i) = -25i^2\)
Now, substitute \(i^2 = -1\):
\(-25i^2 = -25(-1) = 25\)
Now, combine all the terms:
\[ 100 + 50i - 50i + 25 = 100 + 25 = 125 \]
So, the final result of \((-10 + 5i)(-10 - 5i)\) is:
\(\boxed{125}\)
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