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

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

Evaluate

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

Solution Steps

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

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

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

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

To find the opposite of 2a\left(-b\right)+4a^{2}+b\left(-b\right)+2ba, find the opposite of each term.

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

Multiply -2 and -1 to get 2.

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

Combine -2ab and 2ab to get 0.

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

Combine a^{2} and -4a^{2} to get -3a^{2}.

$$-3a^{2}+b^{2}-b\left(-b\right)-2ba$$

Multiply -1 and -1 to get 1.

$$-3a^{2}+b^{2}+bb-2ba$$

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

$$-3a^{2}+b^{2}+b^{2}-2ba$$

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

$$-3a^{2}+2b^{2}-2ba$$

Expand

$$2b^{2}-2ab-3a^{2}$$

Solution Steps

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

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

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

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

To find the opposite of 2a\left(-b\right)+4a^{2}+b\left(-b\right)+2ba, find the opposite of each term.

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

Multiply -2 and -1 to get 2.

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

Combine -2ab and 2ab to get 0.

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

Combine a^{2} and -4a^{2} to get -3a^{2}.

$$-3a^{2}+b^{2}-b\left(-b\right)-2ba$$

Multiply -1 and -1 to get 1.

$$-3a^{2}+b^{2}+bb-2ba$$

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

$$-3a^{2}+b^{2}+b^{2}-2ba$$

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

$$-3a^{2}+2b^{2}-2ba$$