Solve -(1-a)(1+a)(1+a^2)quad(5)(3a-2)(2-3a)-(2a-3)(3-2a)

$$- ( 1 - a ) ( 1 + a ) ( 1 + a ^ { 2 } ) \quad ( 5 ) ( 3 a - 2 ) ( 2 - 3 a ) - ( 2 a - 3 ) ( 3 - 2 a )$$

Evaluate

$$29-72a+49a^{2}-20a^{4}+60a^{5}-45a^{6}$$

Solution Steps

To find the opposite of 1-a, find the opposite of each term.

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

Use the distributive property to multiply -1+a by 1+a and combine like terms.

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

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

$$\left(-1+a^{4}\right)\times 5\left(3a-2\right)\left(2-3a\right)-\left(2a-3\right)\left(3-2a\right)$$

Use the distributive property to multiply -1+a^{4} by 5.

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

Use the distributive property to multiply -5+5a^{4} by 3a-2.

$$\left(-15a+10+15a^{5}-10a^{4}\right)\left(2-3a\right)-\left(2a-3\right)\left(3-2a\right)$$

Use the distributive property to multiply -15a+10+15a^{5}-10a^{4} by 2-3a and combine like terms.

$$-60a+45a^{2}+20+60a^{5}-45a^{6}-20a^{4}-\left(2a-3\right)\left(3-2a\right)$$

Use the distributive property to multiply 2a-3 by 3-2a and combine like terms.

$$-60a+45a^{2}+20+60a^{5}-45a^{6}-20a^{4}-\left(12a-4a^{2}-9\right)$$

To find the opposite of 12a-4a^{2}-9, find the opposite of each term.

$$-60a+45a^{2}+20+60a^{5}-45a^{6}-20a^{4}-12a+4a^{2}+9$$

Combine -60a and -12a to get -72a.

$$-72a+45a^{2}+20+60a^{5}-45a^{6}-20a^{4}+4a^{2}+9$$

Combine 45a^{2} and 4a^{2} to get 49a^{2}.

$$-72a+49a^{2}+20+60a^{5}-45a^{6}-20a^{4}+9$$

Add 20 and 9 to get 29.

$$-72a+49a^{2}+29+60a^{5}-45a^{6}-20a^{4}$$

Expand

$$29-72a+49a^{2}-20a^{4}+60a^{5}-45a^{6}$$

Solution Steps

To find the opposite of 1-a, find the opposite of each term.

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

Use the distributive property to multiply -1+a by 1+a and combine like terms.

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

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

$$\left(-1+a^{4}\right)\times 5\left(3a-2\right)\left(2-3a\right)-\left(2a-3\right)\left(3-2a\right)$$

Use the distributive property to multiply -1+a^{4} by 5.

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

Use the distributive property to multiply -5+5a^{4} by 3a-2.

$$\left(-15a+10+15a^{5}-10a^{4}\right)\left(2-3a\right)-\left(2a-3\right)\left(3-2a\right)$$

Use the distributive property to multiply -15a+10+15a^{5}-10a^{4} by 2-3a and combine like terms.

$$-60a+45a^{2}+20+60a^{5}-45a^{6}-20a^{4}-\left(2a-3\right)\left(3-2a\right)$$

Use the distributive property to multiply 2a-3 by 3-2a and combine like terms.

$$-60a+45a^{2}+20+60a^{5}-45a^{6}-20a^{4}-\left(12a-4a^{2}-9\right)$$

To find the opposite of 12a-4a^{2}-9, find the opposite of each term.

$$-60a+45a^{2}+20+60a^{5}-45a^{6}-20a^{4}-12a+4a^{2}+9$$

Combine -60a and -12a to get -72a.

$$-72a+45a^{2}+20+60a^{5}-45a^{6}-20a^{4}+4a^{2}+9$$

Combine 45a^{2} and 4a^{2} to get 49a^{2}.

$$-72a+49a^{2}+20+60a^{5}-45a^{6}-20a^{4}+9$$

Add 20 and 9 to get 29.

$$-72a+49a^{2}+29+60a^{5}-45a^{6}-20a^{4}$$