Simplify (-5-2i)(3+7i)

$(- 5 - 2 i) (3 + 7 i)$

Step 1

Expand $(- 5 - 2 i) (3 + 7 i)$ using the FOIL Method.

Apply the distributive property.

$- 5 (3 + 7 i) - 2 i (3 + 7 i)$

Apply the distributive property.

$- 5 \cdot 3 - 5 (7 i) - 2 i (3 + 7 i)$

Apply the distributive property.

$- 5 \cdot 3 - 5 (7 i) - 2 i \cdot 3 - 2 i (7 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $- 5$ by $3$ .

$- 15 - 5 (7 i) - 2 i \cdot 3 - 2 i (7 i)$

Multiply $7$ by $- 5$ .

$- 15 - 35 i - 2 i \cdot 3 - 2 i (7 i)$

Multiply $3$ by $- 2$ .

$- 15 - 35 i - 6 i - 2 i (7 i)$

Multiply $- 2 i (7 i)$ .

Multiply $7$ by $- 2$ .

$- 15 - 35 i - 6 i - 14 i i$

Raise $i$ to the power of $1$ .

$- 15 - 35 i - 6 i - 14 (i^{1} i)$

Raise $i$ to the power of $1$ .

$- 15 - 35 i - 6 i - 14 (i^{1} i^{1})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 15 - 35 i - 6 i - 14 i^{1 + 1}$

Add $1$ and $1$ .

$- 15 - 35 i - 6 i - 14 i^{2}$

Rewrite $i^{2}$ as $- 1$ .

$- 15 - 35 i - 6 i - 14 \cdot - 1$

Multiply $- 14$ by $- 1$ .

$- 15 - 35 i - 6 i + 14$

Add $- 15$ and $14$ .

$- 1 - 35 i - 6 i$

Subtract $6 i$ from $- 35 i$ .

$- 1 - 41 i$