Simplify (4-5xi)(1+i)-(4-5xi)(5+i)

$(4 - 5 x i) (1 + i) - (4 - 5 x i) (5 + i)$

Step 1

Simplify each term.

Expand $(4 - 5 x i) (1 + i)$ using the FOIL Method.

Apply the distributive property.

$4 (1 + i) - 5 x i (1 + i) - (4 - 5 x i) (5 + i)$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

Multiply $4$ by $1$ .

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

Multiply $- 5$ by $1$ .

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

Multiply $- 5 x i i$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 1$ by $- 5$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $4$ .

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

Multiply $- 5$ by $- 1$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

$4 + 4 i - 5 x i + 5 x - 4 \cdot 5 - 4 i + 5 x i \cdot 5 + 5 x i i$

Simplify each term.

Multiply $- 4$ by $5$ .

$4 + 4 i - 5 x i + 5 x - 20 - 4 i + 5 x i \cdot 5 + 5 x i i$

Multiply $5$ by $5$ .

$4 + 4 i - 5 x i + 5 x - 20 - 4 i + 25 x i + 5 x i i$

Multiply $5 x i i$ .

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

$4 + 4 i - 5 x i + 5 x - 20 - 4 i + 25 x i + 5 x (i^{1} i)$

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

$4 + 4 i - 5 x i + 5 x - 20 - 4 i + 25 x i + 5 x (i^{1} i^{1})$

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

$4 + 4 i - 5 x i + 5 x - 20 - 4 i + 25 x i + 5 x i^{1 + 1}$

Add $1$ and $1$ .

$4 + 4 i - 5 x i + 5 x - 20 - 4 i + 25 x i + 5 x i^{2}$

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

$4 + 4 i - 5 x i + 5 x - 20 - 4 i + 25 x i + 5 x \cdot - 1$

Multiply $- 1$ by $5$ .

$4 + 4 i - 5 x i + 5 x - 20 - 4 i + 25 x i - 5 x$

Step 2

Simplify by adding terms.

Combine the opposite terms in $4 + 4 i - 5 x i + 5 x - 20 - 4 i + 25 x i - 5 x$ .

Subtract $4 i$ from $4 i$ .

$4 - 5 x i + 5 x - 20 + 0 + 25 x i - 5 x$

Add $4 - 5 x i + 5 x - 20$ and $0$ .

$4 - 5 x i + 5 x - 20 + 25 x i - 5 x$

Subtract $5 x$ from $5 x$ .

$4 - 5 x i - 20 + 25 x i + 0$

Add $4 - 5 x i - 20 + 25 x i$ and $0$ .

$4 - 5 x i - 20 + 25 x i$

Subtract $20$ from $4$ .

$- 5 x i - 16 + 25 x i$

Add $- 5 x i$ and $25 x i$ .

$20 x i - 16$

Simplify (4-5xi)(1+i)-(4-5xi)(5+i)

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