Simplify (4+5i)(x+yi)

$(4 + 5 i) (x + y i)$

Step 1

Expand $(4 + 5 i) (x + y i)$ using the FOIL Method.

Apply the distributive property.

$4 (x + y i) + 5 i (x + y i)$

Apply the distributive property.

$4 x + 4 (y i) + 5 i (x + y i)$

Apply the distributive property.

$4 x + 4 (y i) + 5 i x + 5 i (y i)$

Step 2

Simplify terms.

Simplify each term.

Multiply $5 i (y i)$ .

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

$4 x + 4 y i + 5 i x + 5 (y (i^{1} i))$

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

$4 x + 4 y i + 5 i x + 5 (y (i^{1} i^{1}))$

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

$4 x + 4 y i + 5 i x + 5 (y i^{1 + 1})$

Add $1$ and $1$ .

$4 x + 4 y i + 5 i x + 5 (y i^{2})$

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

$4 x + 4 y i + 5 i x + 5 (y \cdot - 1)$

Move $- 1$ to the left of $y$ .

$4 x + 4 y i + 5 i x + 5 (- 1 \cdot y)$

Rewrite $- 1 y$ as $- y$ .

$4 x + 4 y i + 5 i x + 5 (- y)$

Multiply $- 1$ by $5$ .

$4 x + 4 y i + 5 i x - 5 y$

Reorder.

Move $i$ .

$4 x + 4 y i + 5 x i - 5 y$

Move $4 y i$ .

$4 x + 5 x i + 4 y i - 5 y$