Solve left(5-lambdaright)left(5left(-2-lambdaright)-lambdaleft(-2-lambdaright)right)+0+1left(0+2+lambdaright)

$$\left( 5- \lambda \right) \left( 5 \left( -2- \lambda \right) - \lambda \left( -2- \lambda \right) \right) +0+1 \left( 0+2+ \lambda \right)$$

Evaluate

$$-\left(\lambda -6\right)\left(\lambda -4\right)\left(\lambda +2\right)$$

Solution Steps

Use the distributive property to multiply 5 by -2-\lambda .

$$\left(5-\lambda \right)\left(-10-5\lambda -\lambda \left(-2-\lambda \right)\right)+0+1\left(0+2+\lambda \right)$$

Use the distributive property to multiply \lambda by -2-\lambda .

$$\left(5-\lambda \right)\left(-10-5\lambda -\left(-2\lambda -\lambda ^{2}\right)\right)+0+1\left(0+2+\lambda \right)$$

To find the opposite of -2\lambda -\lambda ^{2}, find the opposite of each term.

$$\left(5-\lambda \right)\left(-10-5\lambda -\left(-2\lambda \right)-\left(-\lambda ^{2}\right)\right)+0+1\left(0+2+\lambda \right)$$

The opposite of -2\lambda is 2\lambda .

$$\left(5-\lambda \right)\left(-10-5\lambda +2\lambda -\left(-\lambda ^{2}\right)\right)+0+1\left(0+2+\lambda \right)$$

The opposite of -\lambda ^{2} is \lambda ^{2}.

$$\left(5-\lambda \right)\left(-10-5\lambda +2\lambda +\lambda ^{2}\right)+0+1\left(0+2+\lambda \right)$$

Combine -5\lambda and 2\lambda to get -3\lambda .

$$\left(5-\lambda \right)\left(-10-3\lambda +\lambda ^{2}\right)+0+1\left(0+2+\lambda \right)$$

Apply the distributive property by multiplying each term of 5-\lambda by each term of -10-3\lambda +\lambda ^{2}.

$$-50-15\lambda +5\lambda ^{2}+10\lambda +3\lambda ^{2}-\lambda ^{3}+0+1\left(0+2+\lambda \right)$$

Combine -15\lambda and 10\lambda to get -5\lambda .

$$-50-5\lambda +5\lambda ^{2}+3\lambda ^{2}-\lambda ^{3}+0+1\left(0+2+\lambda \right)$$

Combine 5\lambda ^{2} and 3\lambda ^{2} to get 8\lambda ^{2}.

$$-50-5\lambda +8\lambda ^{2}-\lambda ^{3}+0+1\left(0+2+\lambda \right)$$

Add -50 and 0 to get -50.

$$-50-5\lambda +8\lambda ^{2}-\lambda ^{3}+1\left(0+2+\lambda \right)$$

Add 0 and 2 to get 2.

$$-50-5\lambda +8\lambda ^{2}-\lambda ^{3}+1\left(2+\lambda \right)$$

Use the distributive property to multiply 1 by 2+\lambda .

$$-50-5\lambda +8\lambda ^{2}-\lambda ^{3}+2+\lambda$$

Add -50 and 2 to get -48.

$$-48-5\lambda +8\lambda ^{2}-\lambda ^{3}+\lambda$$

Combine -5\lambda and \lambda to get -4\lambda .

$$-48-4\lambda +8\lambda ^{2}-\lambda ^{3}$$

Expand

$$-\lambda ^{3}+8\lambda ^{2}-4\lambda -48$$

Solution Steps

Use the distributive property to multiply 5 by -2-\lambda .

$$\left(5-\lambda \right)\left(-10-5\lambda -\lambda \left(-2-\lambda \right)\right)+0+1\left(0+2+\lambda \right)$$

Use the distributive property to multiply \lambda by -2-\lambda .

$$\left(5-\lambda \right)\left(-10-5\lambda -\left(-2\lambda -\lambda ^{2}\right)\right)+0+1\left(0+2+\lambda \right)$$

To find the opposite of -2\lambda -\lambda ^{2}, find the opposite of each term.

$$\left(5-\lambda \right)\left(-10-5\lambda -\left(-2\lambda \right)-\left(-\lambda ^{2}\right)\right)+0+1\left(0+2+\lambda \right)$$

The opposite of -2\lambda is 2\lambda .

$$\left(5-\lambda \right)\left(-10-5\lambda +2\lambda -\left(-\lambda ^{2}\right)\right)+0+1\left(0+2+\lambda \right)$$

The opposite of -\lambda ^{2} is \lambda ^{2}.

$$\left(5-\lambda \right)\left(-10-5\lambda +2\lambda +\lambda ^{2}\right)+0+1\left(0+2+\lambda \right)$$

Combine -5\lambda and 2\lambda to get -3\lambda .

$$\left(5-\lambda \right)\left(-10-3\lambda +\lambda ^{2}\right)+0+1\left(0+2+\lambda \right)$$

Apply the distributive property by multiplying each term of 5-\lambda by each term of -10-3\lambda +\lambda ^{2}.

$$-50-15\lambda +5\lambda ^{2}+10\lambda +3\lambda ^{2}-\lambda ^{3}+0+1\left(0+2+\lambda \right)$$

Combine -15\lambda and 10\lambda to get -5\lambda .

$$-50-5\lambda +5\lambda ^{2}+3\lambda ^{2}-\lambda ^{3}+0+1\left(0+2+\lambda \right)$$

Combine 5\lambda ^{2} and 3\lambda ^{2} to get 8\lambda ^{2}.

$$-50-5\lambda +8\lambda ^{2}-\lambda ^{3}+0+1\left(0+2+\lambda \right)$$

Add -50 and 0 to get -50.

$$-50-5\lambda +8\lambda ^{2}-\lambda ^{3}+1\left(0+2+\lambda \right)$$

Add 0 and 2 to get 2.

$$-50-5\lambda +8\lambda ^{2}-\lambda ^{3}+1\left(2+\lambda \right)$$

Use the distributive property to multiply 1 by 2+\lambda .

$$-50-5\lambda +8\lambda ^{2}-\lambda ^{3}+2+\lambda$$

Add -50 and 2 to get -48.

$$-48-5\lambda +8\lambda ^{2}-\lambda ^{3}+\lambda$$

Combine -5\lambda and \lambda to get -4\lambda .

$$-48-4\lambda +8\lambda ^{2}-\lambda ^{3}$$