Compute the product of the complex numbers (2 + 3i) and (4 − i). What is the result in standard form?

Write down the expression:

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

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

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

Perform each multiplication:
Multiply the real parts:

\$2 \times 4 = 8\$

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

\$2 \times (-i) = -2i\$

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

\$3i \times 4 = 12i\$

Multiply the imaginary parts:

\$3i \times (-i) = -3i^2\$

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

\$-3i^2 = -3(-1) = 3\$

Combine all the results:

\$8 - 2i + 12i + 3\$

Combine the real and imaginary parts:

11 + 10i

TThe result of the product of (2+3i) and (4−i) in standard form is 11+10i.

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

10 - 11i

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

-11 - 10i

This option shows a complex number with a negative real part and a positive imaginary part, which does not correspond to our answer

-10 + 5i

This option matches our computed result, which correctly combines the real part 11 and the imaginary part 10i

11 + 10i

Option

D

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