Solve (x+3)(x+4)/(x+3)x^2-2x-15-4(x+5)/(x-5)(x+3)

$$\frac{ (x+3)(x+4) }{ (x+3) { x }^{ 2 } -2x-15 } - \frac{ 4(x+5) }{ (x-5)(x+3) }$$

Evaluate

$$\frac{120-29x-69x^{2}-27x^{3}-3x^{4}}{\left(x-5\right)\left(x+3\right)\left(x^{3}+3x^{2}-2x-15\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)x^{2}-2x-15 and \left(x-5\right)\left(x+3\right) is \left(x-5\right)\left(x+3\right)\left(\left(x+3\right)x^{2}-2x-15\right). Multiply \frac{\left(x+3\right)\left(x+4\right)}{\left(x+3\right)x^{2}-2x-15} times \frac{\left(x-5\right)\left(x+3\right)}{\left(x-5\right)\left(x+3\right)}. Multiply \frac{4\left(x+5\right)}{\left(x-5\right)\left(x+3\right)} times \frac{\left(x+3\right)x^{2}-2x-15}{\left(x+3\right)x^{2}-2x-15}.

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

Since \frac{\left(x+3\right)\left(x+4\right)\left(x-5\right)\left(x+3\right)}{\left(x-5\right)\left(x+3\right)\left(\left(x+3\right)x^{2}-2x-15\right)} and \frac{4\left(x+5\right)\left(\left(x+3\right)x^{2}-2x-15\right)}{\left(x-5\right)\left(x+3\right)\left(\left(x+3\right)x^{2}-2x-15\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+3\right)\left(x+4\right)\left(x-5\right)\left(x+3\right)-4\left(x+5\right)\left(\left(x+3\right)x^{2}-2x-15\right).

$$\frac{x^{4}+2x^{3}-23x^{2}-60x+3x^{3}+6x^{2}-69x-180-4x^{4}-12x^{3}+8x^{2}+60x-20x^{3}-60x^{2}+40x+300}{\left(x-5\right)\left(x+3\right)\left(\left(x+3\right)x^{2}-2x-15\right)}$$

Combine like terms in x^{4}+2x^{3}-23x^{2}-60x+3x^{3}+6x^{2}-69x-180-4x^{4}-12x^{3}+8x^{2}+60x-20x^{3}-60x^{2}+40x+300.

$$\frac{-3x^{4}-27x^{3}-69x^{2}-29x+120}{\left(x-5\right)\left(x+3\right)\left(\left(x+3\right)x^{2}-2x-15\right)}$$

Expand \left(x-5\right)\left(x+3\right)\left(\left(x+3\right)x^{2}-2x-15\right).

$$\frac{-3x^{4}-27x^{3}-69x^{2}-29x+120}{x^{5}+x^{4}-23x^{3}-56x^{2}+60x+225}$$

Expand

$$\frac{120-29x-69x^{2}-27x^{3}-3x^{4}}{\left(x-5\right)\left(x+3\right)\left(x^{3}+3x^{2}-2x-15\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)x^{2}-2x-15 and \left(x-5\right)\left(x+3\right) is \left(x-5\right)\left(x+3\right)\left(\left(x+3\right)x^{2}-2x-15\right). Multiply \frac{\left(x+3\right)\left(x+4\right)}{\left(x+3\right)x^{2}-2x-15} times \frac{\left(x-5\right)\left(x+3\right)}{\left(x-5\right)\left(x+3\right)}. Multiply \frac{4\left(x+5\right)}{\left(x-5\right)\left(x+3\right)} times \frac{\left(x+3\right)x^{2}-2x-15}{\left(x+3\right)x^{2}-2x-15}.

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

Since \frac{\left(x+3\right)\left(x+4\right)\left(x-5\right)\left(x+3\right)}{\left(x-5\right)\left(x+3\right)\left(\left(x+3\right)x^{2}-2x-15\right)} and \frac{4\left(x+5\right)\left(\left(x+3\right)x^{2}-2x-15\right)}{\left(x-5\right)\left(x+3\right)\left(\left(x+3\right)x^{2}-2x-15\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+3\right)\left(x+4\right)\left(x-5\right)\left(x+3\right)-4\left(x+5\right)\left(\left(x+3\right)x^{2}-2x-15\right).

$$\frac{x^{4}+2x^{3}-23x^{2}-60x+3x^{3}+6x^{2}-69x-180-4x^{4}-12x^{3}+8x^{2}+60x-20x^{3}-60x^{2}+40x+300}{\left(x-5\right)\left(x+3\right)\left(\left(x+3\right)x^{2}-2x-15\right)}$$

Combine like terms in x^{4}+2x^{3}-23x^{2}-60x+3x^{3}+6x^{2}-69x-180-4x^{4}-12x^{3}+8x^{2}+60x-20x^{3}-60x^{2}+40x+300.

$$\frac{-3x^{4}-27x^{3}-69x^{2}-29x+120}{\left(x-5\right)\left(x+3\right)\left(\left(x+3\right)x^{2}-2x-15\right)}$$

Expand \left(x-5\right)\left(x+3\right)\left(\left(x+3\right)x^{2}-2x-15\right).

$$\frac{-3x^{4}-27x^{3}-69x^{2}-29x+120}{x^{5}+x^{4}-23x^{3}-56x^{2}+60x+225}$$