Solve (x-1)(x+2)(x+4)(x+7)/(x-6)(x-3)(x+10)

$$\frac { ( x - 1 ) ( x + 2 ) ( x + 4 ) ( x + 7 ) } { ( x - 6 ) ( x - 3 ) ( x + 10 ) }$$

Expand

$$\frac{x^{4}+12x^{3}+37x^{2}+6x-56}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Short Solution Steps

Apply the distributive property by multiplying each term of x-1 by each term of x+2.

$$\frac{\left(x^{2}+2x-x-2\right)\left(x+4\right)\left(x+7\right)}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Combine 2x and -x to get x.

$$\frac{\left(x^{2}+x-2\right)\left(x+4\right)\left(x+7\right)}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Apply the distributive property by multiplying each term of x^{2}+x-2 by each term of x+4.

$$\frac{\left(x^{3}+4x^{2}+x^{2}+4x-2x-8\right)\left(x+7\right)}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

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

$$\frac{\left(x^{3}+5x^{2}+4x-2x-8\right)\left(x+7\right)}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Combine 4x and -2x to get 2x.

$$\frac{\left(x^{3}+5x^{2}+2x-8\right)\left(x+7\right)}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Apply the distributive property by multiplying each term of x^{3}+5x^{2}+2x-8 by each term of x+7.

$$\frac{x^{4}+7x^{3}+5x^{3}+35x^{2}+2x^{2}+14x-8x-56}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Combine 7x^{3} and 5x^{3} to get 12x^{3}.

$$\frac{x^{4}+12x^{3}+35x^{2}+2x^{2}+14x-8x-56}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Combine 35x^{2} and 2x^{2} to get 37x^{2}.

$$\frac{x^{4}+12x^{3}+37x^{2}+14x-8x-56}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Combine 14x and -8x to get 6x.

$$\frac{x^{4}+12x^{3}+37x^{2}+6x-56}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Apply the distributive property by multiplying each term of x-6 by each term of x-3.

$$\frac{x^{4}+12x^{3}+37x^{2}+6x-56}{\left(x^{2}-3x-6x+18\right)\left(x+10\right)}$$

Combine -3x and -6x to get -9x.

$$\frac{x^{4}+12x^{3}+37x^{2}+6x-56}{\left(x^{2}-9x+18\right)\left(x+10\right)}$$

Apply the distributive property by multiplying each term of x^{2}-9x+18 by each term of x+10.

$$\frac{x^{4}+12x^{3}+37x^{2}+6x-56}{x^{3}+10x^{2}-9x^{2}-90x+18x+180}$$

Combine 10x^{2} and -9x^{2} to get x^{2}.

$$\frac{x^{4}+12x^{3}+37x^{2}+6x-56}{x^{3}+x^{2}-90x+18x+180}$$

Combine -90x and 18x to get -72x.

$$\frac{x^{4}+12x^{3}+37x^{2}+6x-56}{x^{3}+x^{2}-72x+180}$$

Evaluate

$$\frac{\left(x-1\right)\left(x+2\right)\left(x+4\right)\left(x+7\right)}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Short Solution Steps

Apply the distributive property by multiplying each term of x-1 by each term of x+2.

$$\frac{\left(x^{2}+2x-x-2\right)\left(x+4\right)\left(x+7\right)}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Combine 2x and -x to get x.

$$\frac{\left(x^{2}+x-2\right)\left(x+4\right)\left(x+7\right)}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Apply the distributive property by multiplying each term of x^{2}+x-2 by each term of x+4.

$$\frac{\left(x^{3}+4x^{2}+x^{2}+4x-2x-8\right)\left(x+7\right)}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

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

$$\frac{\left(x^{3}+5x^{2}+4x-2x-8\right)\left(x+7\right)}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Combine 4x and -2x to get 2x.

$$\frac{\left(x^{3}+5x^{2}+2x-8\right)\left(x+7\right)}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Apply the distributive property by multiplying each term of x^{3}+5x^{2}+2x-8 by each term of x+7.

$$\frac{x^{4}+7x^{3}+5x^{3}+35x^{2}+2x^{2}+14x-8x-56}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Combine 7x^{3} and 5x^{3} to get 12x^{3}.

$$\frac{x^{4}+12x^{3}+35x^{2}+2x^{2}+14x-8x-56}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Combine 35x^{2} and 2x^{2} to get 37x^{2}.

$$\frac{x^{4}+12x^{3}+37x^{2}+14x-8x-56}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Combine 14x and -8x to get 6x.

$$\frac{x^{4}+12x^{3}+37x^{2}+6x-56}{\left(x-6\right)\left(x-3\right)\left(x+10\right)}$$

Apply the distributive property by multiplying each term of x-6 by each term of x-3.

$$\frac{x^{4}+12x^{3}+37x^{2}+6x-56}{\left(x^{2}-3x-6x+18\right)\left(x+10\right)}$$

Combine -3x and -6x to get -9x.

$$\frac{x^{4}+12x^{3}+37x^{2}+6x-56}{\left(x^{2}-9x+18\right)\left(x+10\right)}$$

Apply the distributive property by multiplying each term of x^{2}-9x+18 by each term of x+10.

$$\frac{x^{4}+12x^{3}+37x^{2}+6x-56}{x^{3}+10x^{2}-9x^{2}-90x+18x+180}$$

Combine 10x^{2} and -9x^{2} to get x^{2}.

$$\frac{x^{4}+12x^{3}+37x^{2}+6x-56}{x^{3}+x^{2}-90x+18x+180}$$

Combine -90x and 18x to get -72x.

$$\frac{x^{4}+12x^{3}+37x^{2}+6x-56}{x^{3}+x^{2}-72x+180}$$