What do you get when multiplying (1 + 2i) by (3 + 4i)? Express your answer as a complex number.

Write down the expression:

\$(1 + 2i) \times (3 + 4i)\$

Apply the distributive property (FOIL method) to expand the product:

\$(1 \times 3) + (1 \times (4i)) + (2i \times 3) + (2i \times (4i))\$

Perform each multiplication:
Multiply the real parts:

\$1 \times 3 = 3\$

Multiply the real part of the first number by the imaginary part of the second number:

\$1 \times (4i) = 4i\$

Multiply the imaginary part of the first number by the real part of the second number:

\$2i \times 3 = 6i\$

Multiply the imaginary parts:

\$2i \times 4i = 8i^2\$
Recall that \$i^2 = -1\$, so:
\$8i^2 = 8(-1) = -8\$

Combine all the results:

3 + 4i + 6i - 8

Combine the real and imaginary parts:

−5 + 10i

The result of multiplying (1+2i) by (3+4i) is −5+10i.

This option represents a complex number with a positive real part and a negative imaginary part, which does not match the result of our computation

5 - 10i

This option shows a complex number with a negative real part and a positive imaginary part, which is not consistent with our computed result

-15 + 5i

This option represents a complex number with a positive real part and a negative imaginary part, which does not match our answer

15 - 5i

This option correctly matches our computed result, which combines the real part -5 and the imaginary part 10i

-5 + 10i i

Option

D

Can you summarize what you’ve understood in the above steps?