Solve (x-1)(x-5)(x-5)-4(x-5)-(x-5)=

$$(x-1)(x-5)(x-5)-4(x-5)-(x-5)=$$

Evaluate

$$x\left(x-6\right)\left(x-5\right)$$

Solution Steps

Multiply x-5 and x-5 to get \left(x-5\right)^{2}.

$$\left(x-1\right)\left(x-5\right)^{2}-4\left(x-5\right)-\left(x-5\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$\left(x-1\right)\left(x^{2}-10x+25\right)-4\left(x-5\right)-\left(x-5\right)$$

Apply the distributive property by multiplying each term of x-1 by each term of x^{2}-10x+25.

$$x^{3}-10x^{2}+25x-x^{2}+10x-25-4\left(x-5\right)-\left(x-5\right)$$

Combine -10x^{2} and -x^{2} to get -11x^{2}.

$$x^{3}-11x^{2}+25x+10x-25-4\left(x-5\right)-\left(x-5\right)$$

Combine 25x and 10x to get 35x.

$$x^{3}-11x^{2}+35x-25-4\left(x-5\right)-\left(x-5\right)$$

Use the distributive property to multiply -4 by x-5.

$$x^{3}-11x^{2}+35x-25-4x+20-\left(x-5\right)$$

Combine 35x and -4x to get 31x.

$$x^{3}-11x^{2}+31x-25+20-\left(x-5\right)$$

Add -25 and 20 to get -5.

$$x^{3}-11x^{2}+31x-5-\left(x-5\right)$$

To find the opposite of x-5, find the opposite of each term.

$$x^{3}-11x^{2}+31x-5-x-\left(-5\right)$$

The opposite of -5 is 5.

$$x^{3}-11x^{2}+31x-5-x+5$$

Combine 31x and -x to get 30x.

$$x^{3}-11x^{2}+30x-5+5$$

Add -5 and 5 to get 0.

$$x^{3}-11x^{2}+30x$$

Expand

$$x^{3}-11x^{2}+30x$$

Solution Steps

Multiply x-5 and x-5 to get \left(x-5\right)^{2}.

$$\left(x-1\right)\left(x-5\right)^{2}-4\left(x-5\right)-\left(x-5\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$\left(x-1\right)\left(x^{2}-10x+25\right)-4\left(x-5\right)-\left(x-5\right)$$

Apply the distributive property by multiplying each term of x-1 by each term of x^{2}-10x+25.

$$x^{3}-10x^{2}+25x-x^{2}+10x-25-4\left(x-5\right)-\left(x-5\right)$$

Combine -10x^{2} and -x^{2} to get -11x^{2}.

$$x^{3}-11x^{2}+25x+10x-25-4\left(x-5\right)-\left(x-5\right)$$

Combine 25x and 10x to get 35x.

$$x^{3}-11x^{2}+35x-25-4\left(x-5\right)-\left(x-5\right)$$

Use the distributive property to multiply -4 by x-5.

$$x^{3}-11x^{2}+35x-25-4x+20-\left(x-5\right)$$

Combine 35x and -4x to get 31x.

$$x^{3}-11x^{2}+31x-25+20-\left(x-5\right)$$

Add -25 and 20 to get -5.

$$x^{3}-11x^{2}+31x-5-\left(x-5\right)$$

To find the opposite of x-5, find the opposite of each term.

$$x^{3}-11x^{2}+31x-5-x-\left(-5\right)$$

The opposite of -5 is 5.

$$x^{3}-11x^{2}+31x-5-x+5$$

Combine 31x and -x to get 30x.

$$x^{3}-11x^{2}+30x-5+5$$

Add -5 and 5 to get 0.

$$x^{3}-11x^{2}+30x$$