Solve x^4+5x^3-14x^2-40x+48/x^4-4{x^3-5{x}^2+36x-36}7x+21/{x^2+2x+4}

$$\frac{ { x }^{ 4 } +5 { x }^{ 3 } -14 { x }^{ 2 } -40x+48 }{ { x }^{ 4 } -4 { x }^{ 3 } -5 { x }^{ 2 } +36x-36 } \frac{ 7x+21 }{ { x }^{ 2 } +2x+4 }$$

Evaluate

$$\frac{7\left(x-1\right)\left(x+6\right)\left(x^{2}-8\right)}{\left(x-3\right)\left(x-2\right)^{2}\left(x^{2}+2x+4\right)}$$

Short Solution Steps

Multiply \frac{x^{4}+5x^{3}-14x^{2}-40x+48}{x^{4}-4x^{3}-5x^{2}+36x-36} times \frac{7x+21}{x^{2}+2x+4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x^{4}+5x^{3}-14x^{2}-40x+48\right)\left(7x+21\right)}{\left(x^{4}-4x^{3}-5x^{2}+36x-36\right)\left(x^{2}+2x+4\right)}$$

Factor the expressions that are not already factored.

$$\frac{7\left(x-1\right)\left(x+3\right)\left(x+6\right)\left(x^{2}-8\right)}{\left(x-3\right)\left(x+3\right)\left(x-2\right)^{2}\left(x^{2}+2x+4\right)}$$

Cancel out x+3 in both numerator and denominator.

$$\frac{7\left(x-1\right)\left(x+6\right)\left(x^{2}-8\right)}{\left(x-3\right)\left(x-2\right)^{2}\left(x^{2}+2x+4\right)}$$

Expand the expression.

$$\frac{7x^{4}+35x^{3}-98x^{2}-280x+336}{x^{5}-5x^{4}+6x^{3}-8x^{2}+40x-48}$$

Expand

$$\frac{7\left(x^{4}+5x^{3}-14x^{2}-40x+48\right)}{\left(x^{2}+2x+4\right)\left(x^{3}-7x^{2}+16x-12\right)}$$

Short Solution Steps

Multiply \frac{x^{4}+5x^{3}-14x^{2}-40x+48}{x^{4}-4x^{3}-5x^{2}+36x-36} times \frac{7x+21}{x^{2}+2x+4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x^{4}+5x^{3}-14x^{2}-40x+48\right)\left(7x+21\right)}{\left(x^{4}-4x^{3}-5x^{2}+36x-36\right)\left(x^{2}+2x+4\right)}$$

Factor the expressions that are not already factored.

$$\frac{7\left(x-1\right)\left(x+3\right)\left(x+6\right)\left(x^{2}-8\right)}{\left(x-3\right)\left(x+3\right)\left(x-2\right)^{2}\left(x^{2}+2x+4\right)}$$

Cancel out x+3 in both numerator and denominator.

$$\frac{7\left(x-1\right)\left(x+6\right)\left(x^{2}-8\right)}{\left(x-3\right)\left(x-2\right)^{2}\left(x^{2}+2x+4\right)}$$

Expand the expression.

$$\frac{7x^{4}+35x^{3}-98x^{2}-280x+336}{x^{5}-5x^{4}+6x^{3}-8x^{2}+40x-48}$$