To solve this problem, we need to add the two given 32-bit binary numbers using the two's complement system and then express the result in decimal. Here are the steps:

Step 1: Analyze the Binary Numbers

The given binary numbers are:

• Binary 1:
  $1111\ 0000\ 0001\ 1100\ 1110\ 0001\ 0011\ 1101$
• Binary 2:
  $1100\ 1110\ 1000\ 0011\ 0010\ 0010\ 0001\ 0111$

Step 2: Determine if the Numbers are Negative or Positive

In two's complement, the most significant bit (MSB) indicates the sign of the number:

• If the MSB is 1, the number is negative.

• If the MSB is 0, the number is positive.

• Binary 1 has an MSB of 1, so it is negative.

• Binary 2 also has an MSB of 1, so it is also negative.

Step 3: Convert the Numbers to Decimal

First, we convert each binary number to its absolute value in decimal, then apply the sign.

Binary 1:

1. Invert the bits (flip 0s to 1s and 1s to 0s): $0000\ 1111\ 1110\ 0011\ 0001\ 1110\ 1100\ 0010$
2. Add 1: $0000\ 1111\ 1110\ 0011\ 0001\ 1110\ 1100\ 0011$
3. Convert to decimal:
   • $0 \times 2^{31} + 0 \times 2^{30} + \cdots + 1 \times 2^4 + 1 \times 2^1 + 1 \times 2^0$
   • Decimal value: $267401411$
   • Since the original number was negative, Binary 1 is $-267401411$.

Binary 2:

1. Invert the bits: $0011\ 0001\ 0111\ 1100\ 1101\ 1101\ 1110\ 1000$
2. Add 1: $0011\ 0001\ 0111\ 1100\ 1101\ 1101\ 1110\ 1001$
3. Convert to decimal:
   • Decimal value: $833731305$
   • Since the original number was negative, Binary 2 is $-833731305$.

Step 4: Add the Two Numbers

• Binary 1: $-267401411$
• Binary 2: $-833731305$

Adding these: $-267401411 + (-833731305) = -1101132716$

Step 5: Express the Result in Decimal

The final result of adding the two binary numbers is $\boxed{-1101132716}$ in decimal.

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. What happens if you add two negative numbers in two's complement?
2. How do you determine if an overflow occurs in two's complement addition?
3. Can you explain how to subtract two binary numbers directly using two's complement?
4. How does two's complement handle large negative numbers?
5. How does the process change if the computer uses 64-bit words instead of 32-bit?

Tip: When adding two negative numbers in two's complement, the result will also be negative unless an overflow occurs, in which case it might wrap around to a positive number.