Solve 4/x^2+3x-18+3/x^2+4x-21/frac{4{x^2+13x+42}+3/x^2+2x-15}

$$\frac { \frac { 4 } { x ^ { 2 } + 3 x - 18 } + \frac { 3 } { x ^ { 2 } + 4 x - 21 } } { \frac { 4 } { x ^ { 2 } + 13 x + 42 } + \frac { 3 } { x ^ { 2 } + 2 x - 15 } }$$

Evaluate

$$\frac{\left(x+5\right)\left(7x+46\right)}{\left(x+2\right)\left(7x+33\right)}$$

Short Solution Steps

Factor x^{2}+3x-18. Factor x^{2}+4x-21.

$$\frac{\frac{4}{\left(x-3\right)\left(x+6\right)}+\frac{3}{\left(x-3\right)\left(x+7\right)}}{\frac{4}{x^{2}+13x+42}+\frac{3}{x^{2}+2x-15}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+6\right) and \left(x-3\right)\left(x+7\right) is \left(x-3\right)\left(x+6\right)\left(x+7\right). Multiply \frac{4}{\left(x-3\right)\left(x+6\right)} times \frac{x+7}{x+7}. Multiply \frac{3}{\left(x-3\right)\left(x+7\right)} times \frac{x+6}{x+6}.

$$\frac{\frac{4\left(x+7\right)}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}+\frac{3\left(x+6\right)}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4}{x^{2}+13x+42}+\frac{3}{x^{2}+2x-15}}$$

Since \frac{4\left(x+7\right)}{\left(x-3\right)\left(x+6\right)\left(x+7\right)} and \frac{3\left(x+6\right)}{\left(x-3\right)\left(x+6\right)\left(x+7\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{4\left(x+7\right)+3\left(x+6\right)}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4}{x^{2}+13x+42}+\frac{3}{x^{2}+2x-15}}$$

Do the multiplications in 4\left(x+7\right)+3\left(x+6\right).

$$\frac{\frac{4x+28+3x+18}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4}{x^{2}+13x+42}+\frac{3}{x^{2}+2x-15}}$$

Combine like terms in 4x+28+3x+18.

$$\frac{\frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4}{x^{2}+13x+42}+\frac{3}{x^{2}+2x-15}}$$

Factor x^{2}+13x+42. Factor x^{2}+2x-15.

$$\frac{\frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4}{\left(x+6\right)\left(x+7\right)}+\frac{3}{\left(x-3\right)\left(x+5\right)}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+6\right)\left(x+7\right) and \left(x-3\right)\left(x+5\right) is \left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right). Multiply \frac{4}{\left(x+6\right)\left(x+7\right)} times \frac{\left(x-3\right)\left(x+5\right)}{\left(x-3\right)\left(x+5\right)}. Multiply \frac{3}{\left(x-3\right)\left(x+5\right)} times \frac{\left(x+6\right)\left(x+7\right)}{\left(x+6\right)\left(x+7\right)}.

$$\frac{\frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4\left(x-3\right)\left(x+5\right)}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}+\frac{3\left(x+6\right)\left(x+7\right)}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}}$$

Since \frac{4\left(x-3\right)\left(x+5\right)}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)} and \frac{3\left(x+6\right)\left(x+7\right)}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4\left(x-3\right)\left(x+5\right)+3\left(x+6\right)\left(x+7\right)}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}}$$

Do the multiplications in 4\left(x-3\right)\left(x+5\right)+3\left(x+6\right)\left(x+7\right).

$$\frac{\frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4x^{2}+20x-12x-60+3x^{2}+21x+18x+126}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}}$$

Combine like terms in 4x^{2}+20x-12x-60+3x^{2}+21x+18x+126.

$$\frac{\frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{7x^{2}+47x+66}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}}$$

Divide \frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)} by \frac{7x^{2}+47x+66}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)} by multiplying \frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)} by the reciprocal of \frac{7x^{2}+47x+66}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}.

$$\frac{\left(7x+46\right)\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}{\left(x-3\right)\left(x+6\right)\left(x+7\right)\left(7x^{2}+47x+66\right)}$$

Cancel out \left(x-3\right)\left(x+6\right)\left(x+7\right) in both numerator and denominator.

$$\frac{\left(x+5\right)\left(7x+46\right)}{7x^{2}+47x+66}$$

Use the distributive property to multiply x+5 by 7x+46 and combine like terms.

$$\frac{7x^{2}+81x+230}{7x^{2}+47x+66}$$

Expand

$$\frac{7x^{2}+81x+230}{7x^{2}+47x+66}$$

Short Solution Steps

Factor x^{2}+3x-18. Factor x^{2}+4x-21.

$$\frac{\frac{4}{\left(x-3\right)\left(x+6\right)}+\frac{3}{\left(x-3\right)\left(x+7\right)}}{\frac{4}{x^{2}+13x+42}+\frac{3}{x^{2}+2x-15}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+6\right) and \left(x-3\right)\left(x+7\right) is \left(x-3\right)\left(x+6\right)\left(x+7\right). Multiply \frac{4}{\left(x-3\right)\left(x+6\right)} times \frac{x+7}{x+7}. Multiply \frac{3}{\left(x-3\right)\left(x+7\right)} times \frac{x+6}{x+6}.

$$\frac{\frac{4\left(x+7\right)}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}+\frac{3\left(x+6\right)}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4}{x^{2}+13x+42}+\frac{3}{x^{2}+2x-15}}$$

Since \frac{4\left(x+7\right)}{\left(x-3\right)\left(x+6\right)\left(x+7\right)} and \frac{3\left(x+6\right)}{\left(x-3\right)\left(x+6\right)\left(x+7\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{4\left(x+7\right)+3\left(x+6\right)}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4}{x^{2}+13x+42}+\frac{3}{x^{2}+2x-15}}$$

Do the multiplications in 4\left(x+7\right)+3\left(x+6\right).

$$\frac{\frac{4x+28+3x+18}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4}{x^{2}+13x+42}+\frac{3}{x^{2}+2x-15}}$$

Combine like terms in 4x+28+3x+18.

$$\frac{\frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4}{x^{2}+13x+42}+\frac{3}{x^{2}+2x-15}}$$

Factor x^{2}+13x+42. Factor x^{2}+2x-15.

$$\frac{\frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4}{\left(x+6\right)\left(x+7\right)}+\frac{3}{\left(x-3\right)\left(x+5\right)}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+6\right)\left(x+7\right) and \left(x-3\right)\left(x+5\right) is \left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right). Multiply \frac{4}{\left(x+6\right)\left(x+7\right)} times \frac{\left(x-3\right)\left(x+5\right)}{\left(x-3\right)\left(x+5\right)}. Multiply \frac{3}{\left(x-3\right)\left(x+5\right)} times \frac{\left(x+6\right)\left(x+7\right)}{\left(x+6\right)\left(x+7\right)}.

$$\frac{\frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4\left(x-3\right)\left(x+5\right)}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}+\frac{3\left(x+6\right)\left(x+7\right)}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}}$$

Since \frac{4\left(x-3\right)\left(x+5\right)}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)} and \frac{3\left(x+6\right)\left(x+7\right)}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4\left(x-3\right)\left(x+5\right)+3\left(x+6\right)\left(x+7\right)}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}}$$

Do the multiplications in 4\left(x-3\right)\left(x+5\right)+3\left(x+6\right)\left(x+7\right).

$$\frac{\frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{4x^{2}+20x-12x-60+3x^{2}+21x+18x+126}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}}$$

Combine like terms in 4x^{2}+20x-12x-60+3x^{2}+21x+18x+126.

$$\frac{\frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)}}{\frac{7x^{2}+47x+66}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}}$$

Divide \frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)} by \frac{7x^{2}+47x+66}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)} by multiplying \frac{7x+46}{\left(x-3\right)\left(x+6\right)\left(x+7\right)} by the reciprocal of \frac{7x^{2}+47x+66}{\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}.

$$\frac{\left(7x+46\right)\left(x-3\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}{\left(x-3\right)\left(x+6\right)\left(x+7\right)\left(7x^{2}+47x+66\right)}$$

Cancel out \left(x-3\right)\left(x+6\right)\left(x+7\right) in both numerator and denominator.

$$\frac{\left(x+5\right)\left(7x+46\right)}{7x^{2}+47x+66}$$

Use the distributive property to multiply x+5 by 7x+46 and combine like terms.

$$\frac{7x^{2}+81x+230}{7x^{2}+47x+66}$$