Solve (a-2b)*[(a+b)*(a-b)-2a*(a-1)^2-2*(-a^3-a)]+a*(10ab-5a^2+b^2)

$$( a - 2 b ) \cdot [ ( a + b ) \cdot ( a - b ) - 2 a \cdot ( a - 1 ) ^ { 2 } - 2 \cdot ( - a ^ { 3 } - a ) ] + a \cdot ( 10 a b - 5 a ^ { 2 } + b ^ { 2 } )$$

Evaluate

$$2b^{3}$$

Solution Steps

Consider \left(a+b\right)\left(a-b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(a-2b\right)\left(a^{2}-b^{2}-2a\left(a-1\right)^{2}-2\left(-a^{3}-a\right)\right)+a\left(10ab-5a^{2}+b^{2}\right)$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-1\right)^{2}.

$$\left(a-2b\right)\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)-2\left(-a^{3}-a\right)\right)+a\left(10ab-5a^{2}+b^{2}\right)$$

Use the distributive property to multiply -2 by -a^{3}-a.

$$\left(a-2b\right)\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)-2\left(-a^{3}\right)+2a\right)+a\left(10ab-5a^{2}+b^{2}\right)$$

Multiply -2 and -1 to get 2.

$$\left(a-2b\right)\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)+2a^{3}+2a\right)+a\left(10ab-5a^{2}+b^{2}\right)$$

Use the distributive property to multiply a-2b by a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)+2a^{3}+2a.

$$a\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)+2a^{4}+2a^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+a\left(10ab-5a^{2}+b^{2}\right)$$

Use the distributive property to multiply a by 10ab-5a^{2}+b^{2}.

$$a\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)+2a^{4}+2a^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Use the distributive property to multiply -2a by a^{2}-2a+1.

$$a\left(a^{2}-b^{2}-2a^{3}+4a^{2}-2a\right)+2a^{4}+2a^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Combine a^{2} and 4a^{2} to get 5a^{2}.

$$a\left(5a^{2}-b^{2}-2a^{3}-2a\right)+2a^{4}+2a^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Use the distributive property to multiply a by 5a^{2}-b^{2}-2a^{3}-2a.

$$5a^{3}-ab^{2}-2a^{4}-2a^{2}+2a^{4}+2a^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Combine -2a^{4} and 2a^{4} to get 0.

$$5a^{3}-ab^{2}-2a^{2}+2a^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

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

$$5a^{3}-ab^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Use the distributive property to multiply -2a by a^{2}-2a+1.

$$5a^{3}-ab^{2}-2b\left(a^{2}-b^{2}-2a^{3}+4a^{2}-2a\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Combine a^{2} and 4a^{2} to get 5a^{2}.

$$5a^{3}-ab^{2}-2b\left(5a^{2}-b^{2}-2a^{3}-2a\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Use the distributive property to multiply -2b by 5a^{2}-b^{2}-2a^{3}-2a.

$$5a^{3}-ab^{2}-10ba^{2}+2b^{3}+4ba^{3}+4ba-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Combine 4ba^{3} and -4ba^{3} to get 0.

$$5a^{3}-ab^{2}-10ba^{2}+2b^{3}+4ba-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Combine 4ba and -4ba to get 0.

$$5a^{3}-ab^{2}-10ba^{2}+2b^{3}+10ba^{2}-5a^{3}+ab^{2}$$

Combine -10ba^{2} and 10ba^{2} to get 0.

$$5a^{3}-ab^{2}+2b^{3}-5a^{3}+ab^{2}$$

Combine 5a^{3} and -5a^{3} to get 0.

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

Combine -ab^{2} and ab^{2} to get 0.

$$2b^{3}$$

Expand

$$2b^{3}$$

Solution Steps

Consider \left(a+b\right)\left(a-b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(a-2b\right)\left(a^{2}-b^{2}-2a\left(a-1\right)^{2}-2\left(-a^{3}-a\right)\right)+a\left(10ab-5a^{2}+b^{2}\right)$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-1\right)^{2}.

$$\left(a-2b\right)\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)-2\left(-a^{3}-a\right)\right)+a\left(10ab-5a^{2}+b^{2}\right)$$

Use the distributive property to multiply -2 by -a^{3}-a.

$$\left(a-2b\right)\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)-2\left(-a^{3}\right)+2a\right)+a\left(10ab-5a^{2}+b^{2}\right)$$

Multiply -2 and -1 to get 2.

$$\left(a-2b\right)\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)+2a^{3}+2a\right)+a\left(10ab-5a^{2}+b^{2}\right)$$

Use the distributive property to multiply a-2b by a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)+2a^{3}+2a.

$$a\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)+2a^{4}+2a^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+a\left(10ab-5a^{2}+b^{2}\right)$$

Use the distributive property to multiply a by 10ab-5a^{2}+b^{2}.

$$a\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)+2a^{4}+2a^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Use the distributive property to multiply -2a by a^{2}-2a+1.

$$a\left(a^{2}-b^{2}-2a^{3}+4a^{2}-2a\right)+2a^{4}+2a^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Combine a^{2} and 4a^{2} to get 5a^{2}.

$$a\left(5a^{2}-b^{2}-2a^{3}-2a\right)+2a^{4}+2a^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Use the distributive property to multiply a by 5a^{2}-b^{2}-2a^{3}-2a.

$$5a^{3}-ab^{2}-2a^{4}-2a^{2}+2a^{4}+2a^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Combine -2a^{4} and 2a^{4} to get 0.

$$5a^{3}-ab^{2}-2a^{2}+2a^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

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

$$5a^{3}-ab^{2}-2b\left(a^{2}-b^{2}-2a\left(a^{2}-2a+1\right)\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Use the distributive property to multiply -2a by a^{2}-2a+1.

$$5a^{3}-ab^{2}-2b\left(a^{2}-b^{2}-2a^{3}+4a^{2}-2a\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Combine a^{2} and 4a^{2} to get 5a^{2}.

$$5a^{3}-ab^{2}-2b\left(5a^{2}-b^{2}-2a^{3}-2a\right)-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Use the distributive property to multiply -2b by 5a^{2}-b^{2}-2a^{3}-2a.

$$5a^{3}-ab^{2}-10ba^{2}+2b^{3}+4ba^{3}+4ba-4ba^{3}-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Combine 4ba^{3} and -4ba^{3} to get 0.

$$5a^{3}-ab^{2}-10ba^{2}+2b^{3}+4ba-4ba+10ba^{2}-5a^{3}+ab^{2}$$

Combine 4ba and -4ba to get 0.

$$5a^{3}-ab^{2}-10ba^{2}+2b^{3}+10ba^{2}-5a^{3}+ab^{2}$$

Combine -10ba^{2} and 10ba^{2} to get 0.

$$5a^{3}-ab^{2}+2b^{3}-5a^{3}+ab^{2}$$

Combine 5a^{3} and -5a^{3} to get 0.

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

Combine -ab^{2} and ab^{2} to get 0.

$$2b^{3}$$