To determine which expression is equivalent to [tex]\((4 + 7i)(3 + 4i)\)[/tex], let's go through the multiplication of these two complex numbers step by step.
To multiply two complex numbers, we can use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(4 + 7i)(3 + 4i)
\][/tex]
Step 1: Distribute [tex]\(4\)[/tex] in the first term over the second complex number:
[tex]\[
4 \cdot 3 + 4 \cdot 4i = 12 + 16i
\][/tex]
Step 2: Distribute [tex]\(7i\)[/tex] in the first term over the second complex number:
[tex]\[
7i \cdot 3 + 7i \cdot 4i = 21i + 28i^2
\][/tex]
Step 3: Combine the results:
[tex]\[
12 + 16i + 21i + 28i^2
\][/tex]
Step 4: Recall that [tex]\(i^2 = -1\)[/tex], so substitute [tex]\(-1\)[/tex] in for [tex]\(i^2\)[/tex]:
[tex]\[
12 + 16i + 21i + 28(-1)
\][/tex]
Step 5: Simplify the expression:
[tex]\[
12 + 16i + 21i - 28
\][/tex]
Step 6: Combine like terms (real parts together and imaginary parts together):
[tex]\[
(12 - 28) + (16i + 21i) = -16 + 37i
\][/tex]
Therefore, the expression that is equivalent to [tex]\((4 + 7i)(3 + 4i)\)[/tex] is:
[tex]\[
-16 + 37i
\][/tex]
So, the correct option is:
[tex]\[
-16 + 37i
\][/tex]
