Solve 5x/2x^2+5x-3-2x/x^2+6x+9

$$\frac { 5 x } { 2 x ^ { 2 } + 5 x - 3 } - \frac { 2 x } { x ^ { 2 } + 6 x + 9 }$$

Evaluate

$$\frac{x\left(x+17\right)}{\left(2x-1\right)\left(x+3\right)^{2}}$$

Short Solution Steps

Factor 2x^{2}+5x-3. Factor x^{2}+6x+9.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2x-1\right)\left(x+3\right) and \left(x+3\right)^{2} is \left(2x-1\right)\left(x+3\right)^{2}. Multiply \frac{5x}{\left(2x-1\right)\left(x+3\right)} times \frac{x+3}{x+3}. Multiply \frac{2x}{\left(x+3\right)^{2}} times \frac{2x-1}{2x-1}.

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

Since \frac{5x\left(x+3\right)}{\left(2x-1\right)\left(x+3\right)^{2}} and \frac{2x\left(2x-1\right)}{\left(2x-1\right)\left(x+3\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 5x\left(x+3\right)-2x\left(2x-1\right).

$$\frac{5x^{2}+15x-4x^{2}+2x}{\left(2x-1\right)\left(x+3\right)^{2}}$$

Combine like terms in 5x^{2}+15x-4x^{2}+2x.

$$\frac{x^{2}+17x}{\left(2x-1\right)\left(x+3\right)^{2}}$$

Expand \left(2x-1\right)\left(x+3\right)^{2}.

$$\frac{x^{2}+17x}{2x^{3}+11x^{2}+12x-9}$$

Factor

$$\frac{x\left(x+17\right)}{\left(2x-1\right)\left(x+3\right)^{2}}$$