Final answer:
The simplified expression is obtained by distributing and combining like terms. The given expression 2(-36 - 3i) +(5 +2i)(12 - 2i) simplifies to -8 + 8i.
Step-by-step explanation:
To simplify the given expression into the form a+bi, where a and b are rational numbers, we first distribute and combine like terms.
Starting with the expression:
2(-36 - 3i) +(5 +2i)(12 - 2i)
First, distribute the 2 in the expression 2(-36 - 3i):
-72 - 6i
Next, use the FOIL method (First, Outer, Inner, Last) to expand (5 +2i)(12 - 2i):
5 * 12 + 5 * (-2i) + 2i * 12 + 2i * (-2i)
Which simplifies to:
60 - 10i + 24i - 4(i^2)
Since i^2 is -1, this further simplifies to:
60 + 14i + 4
Combine like terms:
64 + 14i
Add this result to the -72 - 6i from the first distribution:
-72 - 6i + 64 + 14i
Combine like terms:
(-72 + 64) + (-6i + 14i)
Which simplifies to:
-8 + 8i
Therefore, the simplified expression in the form a+bi is:
-8 + 8i