Simplify (ab^2+c)(ab^2-c)

$(a b^{2} + c) (a b^{2} - c)$

Step 1

Expand $(a b^{2} + c) (a b^{2} - c)$ using the FOIL Method.

Apply the distributive property.

$a b^{2} (a b^{2} - c) + c (a b^{2} - c)$

Apply the distributive property.

$a b^{2} (a b^{2}) + a b^{2} (- c) + c (a b^{2} - c)$

Apply the distributive property.

$a b^{2} (a b^{2}) + a b^{2} (- c) + c (a b^{2}) + c (- c)$

Step 2

Simplify terms.

Combine the opposite terms in $a b^{2} (a b^{2}) + a b^{2} (- c) + c (a b^{2}) + c (- c)$ .

Reorder the factors in the terms $a b^{2} (- c)$ and $c (a b^{2})$ .

$a b^{2} (a b^{2}) - b^{2} a c + b^{2} a c + c (- c)$

Add $- b^{2} a c$ and $b^{2} a c$ .

$a b^{2} (a b^{2}) + 0 + c (- c)$

Add $a b^{2} (a b^{2})$ and $0$ .

$a b^{2} (a b^{2}) + c (- c)$

Simplify each term.

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$a \cdot a b^{2} b^{2} + c (- c)$

Multiply $a$ by $a$ .

$a^{2} b^{2} b^{2} + c (- c)$

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

Move $b^{2}$ .

$a^{2} (b^{2} b^{2}) + c (- c)$

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

$a^{2} b^{2 + 2} + c (- c)$

Add $2$ and $2$ .

$a^{2} b^{4} + c (- c)$

Rewrite using the commutative property of multiplication.

$a^{2} b^{4} - c \cdot c$

Multiply $c$ by $c$ by adding the exponents.

Move $c$ .

$a^{2} b^{4} - (c \cdot c)$

Multiply $c$ by $c$ .

$a^{2} b^{4} - c^{2}$