Solve x^2+8x-20/x^2+13*+30+x^2+2x-3/x^2+6*+9

$$\frac{ { x }^{ 2 } +8x-20 }{ { x }^{ 2 } +13 \times +30 } + \frac{ { x }^{ 2 } +2x-3 }{ { x }^{ 2 } +6 \times +9 }$$

Evaluate

$$\frac{2x^{4}+10x^{3}+421x^{2}+1212x-2250}{\left(x^{2}+54\right)\left(x^{2}+390\right)}$$

Short Solution Steps

Multiply 13 and 30 to get 390.

$$\frac{x^{2}+8x-20}{x^{2}+390}+\frac{x^{2}+2x-3}{x^{2}+6\times 9}$$

Multiply 6 and 9 to get 54.

$$\frac{x^{2}+8x-20}{x^{2}+390}+\frac{x^{2}+2x-3}{x^{2}+54}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}+390 and x^{2}+54 is \left(x^{2}+54\right)\left(x^{2}+390\right). Multiply \frac{x^{2}+8x-20}{x^{2}+390} times \frac{x^{2}+54}{x^{2}+54}. Multiply \frac{x^{2}+2x-3}{x^{2}+54} times \frac{x^{2}+390}{x^{2}+390}.

$$\frac{\left(x^{2}+8x-20\right)\left(x^{2}+54\right)}{\left(x^{2}+54\right)\left(x^{2}+390\right)}+\frac{\left(x^{2}+2x-3\right)\left(x^{2}+390\right)}{\left(x^{2}+54\right)\left(x^{2}+390\right)}$$

Since \frac{\left(x^{2}+8x-20\right)\left(x^{2}+54\right)}{\left(x^{2}+54\right)\left(x^{2}+390\right)} and \frac{\left(x^{2}+2x-3\right)\left(x^{2}+390\right)}{\left(x^{2}+54\right)\left(x^{2}+390\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x^{2}+8x-20\right)\left(x^{2}+54\right)+\left(x^{2}+2x-3\right)\left(x^{2}+390\right)}{\left(x^{2}+54\right)\left(x^{2}+390\right)}$$

Do the multiplications in \left(x^{2}+8x-20\right)\left(x^{2}+54\right)+\left(x^{2}+2x-3\right)\left(x^{2}+390\right).

$$\frac{x^{4}+54x^{2}+8x^{3}+432x-20x^{2}-1080+x^{4}+390x^{2}+2x^{3}+780x-3x^{2}-1170}{\left(x^{2}+54\right)\left(x^{2}+390\right)}$$

Combine like terms in x^{4}+54x^{2}+8x^{3}+432x-20x^{2}-1080+x^{4}+390x^{2}+2x^{3}+780x-3x^{2}-1170.

$$\frac{2x^{4}+421x^{2}+10x^{3}+1212x-2250}{\left(x^{2}+54\right)\left(x^{2}+390\right)}$$

Expand \left(x^{2}+54\right)\left(x^{2}+390\right).

$$\frac{2x^{4}+421x^{2}+10x^{3}+1212x-2250}{x^{4}+444x^{2}+21060}$$

Expand

$$\frac{2x^{4}+10x^{3}+421x^{2}+1212x-2250}{\left(x^{2}+54\right)\left(x^{2}+390\right)}$$

Short Solution Steps

Multiply 13 and 30 to get 390.

$$\frac{x^{2}+8x-20}{x^{2}+390}+\frac{x^{2}+2x-3}{x^{2}+6\times 9}$$

Multiply 6 and 9 to get 54.

$$\frac{x^{2}+8x-20}{x^{2}+390}+\frac{x^{2}+2x-3}{x^{2}+54}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}+390 and x^{2}+54 is \left(x^{2}+54\right)\left(x^{2}+390\right). Multiply \frac{x^{2}+8x-20}{x^{2}+390} times \frac{x^{2}+54}{x^{2}+54}. Multiply \frac{x^{2}+2x-3}{x^{2}+54} times \frac{x^{2}+390}{x^{2}+390}.

$$\frac{\left(x^{2}+8x-20\right)\left(x^{2}+54\right)}{\left(x^{2}+54\right)\left(x^{2}+390\right)}+\frac{\left(x^{2}+2x-3\right)\left(x^{2}+390\right)}{\left(x^{2}+54\right)\left(x^{2}+390\right)}$$

Since \frac{\left(x^{2}+8x-20\right)\left(x^{2}+54\right)}{\left(x^{2}+54\right)\left(x^{2}+390\right)} and \frac{\left(x^{2}+2x-3\right)\left(x^{2}+390\right)}{\left(x^{2}+54\right)\left(x^{2}+390\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x^{2}+8x-20\right)\left(x^{2}+54\right)+\left(x^{2}+2x-3\right)\left(x^{2}+390\right)}{\left(x^{2}+54\right)\left(x^{2}+390\right)}$$

Do the multiplications in \left(x^{2}+8x-20\right)\left(x^{2}+54\right)+\left(x^{2}+2x-3\right)\left(x^{2}+390\right).

$$\frac{x^{4}+54x^{2}+8x^{3}+432x-20x^{2}-1080+x^{4}+390x^{2}+2x^{3}+780x-3x^{2}-1170}{\left(x^{2}+54\right)\left(x^{2}+390\right)}$$

Combine like terms in x^{4}+54x^{2}+8x^{3}+432x-20x^{2}-1080+x^{4}+390x^{2}+2x^{3}+780x-3x^{2}-1170.

$$\frac{2x^{4}+421x^{2}+10x^{3}+1212x-2250}{\left(x^{2}+54\right)\left(x^{2}+390\right)}$$

Expand \left(x^{2}+54\right)\left(x^{2}+390\right).

$$\frac{2x^{4}+421x^{2}+10x^{3}+1212x-2250}{x^{4}+444x^{2}+21060}$$