Solve (a-4b)^2-2(a+2b)(a-2)

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

Evaluate

$$-2\left(a-2\right)\left(a+2b\right)+\left(a-4b\right)^{2}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-4b\right)^{2}.

$$a^{2}-8ab+16b^{2}-2\left(a+2b\right)\left(a-2\right)$$

Use the distributive property to multiply -2 by a+2b.

$$a^{2}-8ab+16b^{2}+\left(-2a-4b\right)\left(a-2\right)$$

Use the distributive property to multiply -2a-4b by a-2.

$$a^{2}-8ab+16b^{2}-2a^{2}+4a-4ba+8b$$

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

$$-a^{2}-8ab+16b^{2}+4a-4ba+8b$$

Combine -8ab and -4ba to get -12ab.

$$-a^{2}-12ab+16b^{2}+4a+8b$$

Expand

$$8b+16b^{2}+4a-12ab-a^{2}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-4b\right)^{2}.

$$a^{2}-8ab+16b^{2}-2\left(a+2b\right)\left(a-2\right)$$

Use the distributive property to multiply -2 by a+2b.

$$a^{2}-8ab+16b^{2}+\left(-2a-4b\right)\left(a-2\right)$$

Use the distributive property to multiply -2a-4b by a-2.

$$a^{2}-8ab+16b^{2}-2a^{2}+4a-4ba+8b$$

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

$$-a^{2}-8ab+16b^{2}+4a-4ba+8b$$

Combine -8ab and -4ba to get -12ab.

$$-a^{2}-12ab+16b^{2}+4a+8b$$