Find the product. Express your answer in standard form.
$2i(9-4i)(6+5i)$

• A. $8+18i$
• B. $54-20i$
• C. $54-40i$
• D. $-42+148i$

Answer: (D)

Verified step by step guidance

1. First, recognize that the expression involves multiplying three factors: \$2i\(, \)(9 - 4i)\(, and \)(6 + 5i)$. We will multiply them step-by-step.
2. Start by multiplying the two complex binomials \((9 - 4i)\) and \((6 + 5i)\) using the distributive property (FOIL method): \[(9 - 4i)(6 + 5i) = 9 \times 6 + 9 \times 5i - 4i \times 6 - 4i \times 5i\]
3. Simplify each term: \(9 \times 6 = 54\) \(9 \times 5i = 45i\) \(-4i \times 6 = -24i\) \(-4i \times 5i = -20i^2\)
4. Recall that \(i^2 = -1\), so replace \(-20i^2\) with \(-20 \times (-1) = 20\). Now combine all terms: \$54 + 45i - 24i + 20$
5. Combine like terms: Real parts: \$54 + 20 = 74$ Imaginary parts: \$45i - 24i = 21i$ So, \((9 - 4i)(6 + 5i) = 74 + 21i\). Next, multiply this result by \$2i\(: \(2i \times (74 + 21i) = 2i \times 74 + 2i \times 21i = 148i + 42i^2\). Again, replace \)i^2\( with \)-1$ to simplify further.