Solve 3x+5/x-4-7x^2+38x+59/3x^2-5x-28+x+1/3x+7

$$\frac { 3 x + 5 } { x - 4 } - \frac { 7 x ^ { 2 } + 38 x + 59 } { 3 x ^ { 2 } - 5 x - 28 } + \frac { x + 1 } { 3 x + 7 }$$

Evaluate

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Short Solution Steps

Factor 3x^{2}-5x-28.

$$\frac{3x+5}{x-4}-\frac{7x^{2}+38x+59}{\left(x-4\right)\left(3x+7\right)}+\frac{x+1}{3x+7}$$

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

$$\frac{\left(3x+5\right)\left(3x+7\right)}{\left(x-4\right)\left(3x+7\right)}-\frac{7x^{2}+38x+59}{\left(x-4\right)\left(3x+7\right)}+\frac{x+1}{3x+7}$$

Since \frac{\left(3x+5\right)\left(3x+7\right)}{\left(x-4\right)\left(3x+7\right)} and \frac{7x^{2}+38x+59}{\left(x-4\right)\left(3x+7\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3x+5\right)\left(3x+7\right)-\left(7x^{2}+38x+59\right)}{\left(x-4\right)\left(3x+7\right)}+\frac{x+1}{3x+7}$$

Do the multiplications in \left(3x+5\right)\left(3x+7\right)-\left(7x^{2}+38x+59\right).

$$\frac{9x^{2}+21x+15x+35-7x^{2}-38x-59}{\left(x-4\right)\left(3x+7\right)}+\frac{x+1}{3x+7}$$

Combine like terms in 9x^{2}+21x+15x+35-7x^{2}-38x-59.

$$\frac{2x^{2}-2x-24}{\left(x-4\right)\left(3x+7\right)}+\frac{x+1}{3x+7}$$

Factor the expressions that are not already factored in \frac{2x^{2}-2x-24}{\left(x-4\right)\left(3x+7\right)}.

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

Cancel out x-4 in both numerator and denominator.

$$\frac{2\left(x+3\right)}{3x+7}+\frac{x+1}{3x+7}$$

Since \frac{2\left(x+3\right)}{3x+7} and \frac{x+1}{3x+7} have the same denominator, add them by adding their numerators.

$$\frac{2\left(x+3\right)+x+1}{3x+7}$$

Do the multiplications in 2\left(x+3\right)+x+1.

$$\frac{2x+6+x+1}{3x+7}$$

Combine like terms in 2x+6+x+1.

$$\frac{3x+7}{3x+7}$$

Cancel out 3x+7 in both numerator and denominator.

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Factor

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