Solve (ab+bc)^2-(ab-bc)^2

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

Evaluate

$$4acb^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(ab+bc\right)^{2}.

$$a^{2}b^{2}+2abbc+b^{2}c^{2}-\left(ab-bc\right)^{2}$$

Multiply b and b to get b^{2}.

$$a^{2}b^{2}+2ab^{2}c+b^{2}c^{2}-\left(ab-bc\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(ab-bc\right)^{2}.

$$a^{2}b^{2}+2ab^{2}c+b^{2}c^{2}-\left(a^{2}b^{2}-2abbc+b^{2}c^{2}\right)$$

Multiply b and b to get b^{2}.

$$a^{2}b^{2}+2ab^{2}c+b^{2}c^{2}-\left(a^{2}b^{2}-2ab^{2}c+b^{2}c^{2}\right)$$

To find the opposite of a^{2}b^{2}-2ab^{2}c+b^{2}c^{2}, find the opposite of each term.

$$a^{2}b^{2}+2ab^{2}c+b^{2}c^{2}-a^{2}b^{2}+2ab^{2}c-b^{2}c^{2}$$

Combine a^{2}b^{2} and -a^{2}b^{2} to get 0.

$$2ab^{2}c+b^{2}c^{2}+2ab^{2}c-b^{2}c^{2}$$

Combine 2ab^{2}c and 2ab^{2}c to get 4ab^{2}c.

$$4ab^{2}c+b^{2}c^{2}-b^{2}c^{2}$$

Combine b^{2}c^{2} and -b^{2}c^{2} to get 0.

$$4ab^{2}c$$

Expand

$$4acb^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(ab+bc\right)^{2}.

$$a^{2}b^{2}+2abbc+b^{2}c^{2}-\left(ab-bc\right)^{2}$$

Multiply b and b to get b^{2}.

$$a^{2}b^{2}+2ab^{2}c+b^{2}c^{2}-\left(ab-bc\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(ab-bc\right)^{2}.

$$a^{2}b^{2}+2ab^{2}c+b^{2}c^{2}-\left(a^{2}b^{2}-2abbc+b^{2}c^{2}\right)$$

Multiply b and b to get b^{2}.

$$a^{2}b^{2}+2ab^{2}c+b^{2}c^{2}-\left(a^{2}b^{2}-2ab^{2}c+b^{2}c^{2}\right)$$

To find the opposite of a^{2}b^{2}-2ab^{2}c+b^{2}c^{2}, find the opposite of each term.

$$a^{2}b^{2}+2ab^{2}c+b^{2}c^{2}-a^{2}b^{2}+2ab^{2}c-b^{2}c^{2}$$

Combine a^{2}b^{2} and -a^{2}b^{2} to get 0.

$$2ab^{2}c+b^{2}c^{2}+2ab^{2}c-b^{2}c^{2}$$

Combine 2ab^{2}c and 2ab^{2}c to get 4ab^{2}c.

$$4ab^{2}c+b^{2}c^{2}-b^{2}c^{2}$$

Combine b^{2}c^{2} and -b^{2}c^{2} to get 0.

$$4ab^{2}c$$