Expand the Trigonometric Expression (q^2-2q+5)(q^2-5)

$(q^{2} - 2 q + 5) (q^{2} - 5)$

Step 1

Expand $(q^{2} - 2 q + 5) (q^{2} - 5)$ by multiplying each term in the first expression by each term in the second expression.

$q^{2} q^{2} + q^{2} \cdot - 5 - 2 q \cdot q^{2} - 2 q \cdot - 5 + 5 q^{2} + 5 \cdot - 5$

Step 2

Simplify terms.

Combine the opposite terms in $q^{2} q^{2} + q^{2} \cdot - 5 - 2 q \cdot q^{2} - 2 q \cdot - 5 + 5 q^{2} + 5 \cdot - 5$ .

Reorder the factors in the terms $q^{2} \cdot - 5$ and $5 q^{2}$ .

$q^{2} q^{2} - 5 q^{2} - 2 q \cdot q^{2} - 2 q \cdot - 5 + 5 q^{2} + 5 \cdot - 5$

Add $- 5 q^{2}$ and $5 q^{2}$ .

$q^{2} q^{2} - 2 q \cdot q^{2} - 2 q \cdot - 5 + 0 + 5 \cdot - 5$

Add $q^{2} q^{2} - 2 q \cdot q^{2} - 2 q \cdot - 5$ and $0$ .

$q^{2} q^{2} - 2 q \cdot q^{2} - 2 q \cdot - 5 + 5 \cdot - 5$

Simplify each term.

Multiply $q^{2}$ by $q^{2}$ by adding the exponents.

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

$q^{2 + 2} - 2 q \cdot q^{2} - 2 q \cdot - 5 + 5 \cdot - 5$

Add $2$ and $2$ .

$q^{4} - 2 q \cdot q^{2} - 2 q \cdot - 5 + 5 \cdot - 5$

Multiply $q$ by $q^{2}$ by adding the exponents.

Move $q^{2}$ .

$q^{4} - 2 (q^{2} q) - 2 q \cdot - 5 + 5 \cdot - 5$

Multiply $q^{2}$ by $q$ .

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

$q^{4} - 2 (q^{2} q^{1}) - 2 q \cdot - 5 + 5 \cdot - 5$

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

$q^{4} - 2 q^{2 + 1} - 2 q \cdot - 5 + 5 \cdot - 5$

Add $2$ and $1$ .

$q^{4} - 2 q^{3} - 2 q \cdot - 5 + 5 \cdot - 5$

Multiply $- 5$ by $- 2$ .

$q^{4} - 2 q^{3} + 10 q + 5 \cdot - 5$

Multiply $5$ by $- 5$ .

$q^{4} - 2 q^{3} + 10 q - 25$