Solve f8f^2-5/f^2-10f+25-9-f/f+5

$$f \frac{ 8 { f }^{ 2 } -5 }{ { f }^{ 2 } -10f+25 } - \frac{ 9-f }{ f+5 }$$

Evaluate

$$\frac{8f^{4}+41f^{3}-24f^{2}+90f-225}{\left(f+5\right)\left(f-5\right)^{2}}$$

Short Solution Steps

Express f\times \frac{8f^{2}-5}{f^{2}-10f+25} as a single fraction.

$$\frac{f\left(8f^{2}-5\right)}{f^{2}-10f+25}-\frac{9-f}{f+5}$$

Factor f^{2}-10f+25.

$$\frac{f\left(8f^{2}-5\right)}{\left(f-5\right)^{2}}-\frac{9-f}{f+5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(f-5\right)^{2} and f+5 is \left(f+5\right)\left(f-5\right)^{2}. Multiply \frac{f\left(8f^{2}-5\right)}{\left(f-5\right)^{2}} times \frac{f+5}{f+5}. Multiply \frac{9-f}{f+5} times \frac{\left(f-5\right)^{2}}{\left(f-5\right)^{2}}.

$$\frac{f\left(8f^{2}-5\right)\left(f+5\right)}{\left(f+5\right)\left(f-5\right)^{2}}-\frac{\left(9-f\right)\left(f-5\right)^{2}}{\left(f+5\right)\left(f-5\right)^{2}}$$

Since \frac{f\left(8f^{2}-5\right)\left(f+5\right)}{\left(f+5\right)\left(f-5\right)^{2}} and \frac{\left(9-f\right)\left(f-5\right)^{2}}{\left(f+5\right)\left(f-5\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{f\left(8f^{2}-5\right)\left(f+5\right)-\left(9-f\right)\left(f-5\right)^{2}}{\left(f+5\right)\left(f-5\right)^{2}}$$

Do the multiplications in f\left(8f^{2}-5\right)\left(f+5\right)-\left(9-f\right)\left(f-5\right)^{2}.

$$\frac{8f^{4}+40f^{3}-5f^{2}-25f-9f^{2}+90f-225+f^{3}-10f^{2}+25f}{\left(f+5\right)\left(f-5\right)^{2}}$$

Combine like terms in 8f^{4}+40f^{3}-5f^{2}-25f-9f^{2}+90f-225+f^{3}-10f^{2}+25f.

$$\frac{8f^{4}+41f^{3}-24f^{2}+90f-225}{\left(f+5\right)\left(f-5\right)^{2}}$$

Expand \left(f+5\right)\left(f-5\right)^{2}.

$$\frac{8f^{4}+41f^{3}-24f^{2}+90f-225}{f^{3}-5f^{2}-25f+125}$$

Expand

$$\frac{8f^{4}+41f^{3}-24f^{2}+90f-225}{\left(f+5\right)\left(f^{2}-10f+25\right)}$$

Short Solution Steps

Express f\times \frac{8f^{2}-5}{f^{2}-10f+25} as a single fraction.

$$\frac{f\left(8f^{2}-5\right)}{f^{2}-10f+25}-\frac{9-f}{f+5}$$

Factor f^{2}-10f+25.

$$\frac{f\left(8f^{2}-5\right)}{\left(f-5\right)^{2}}-\frac{9-f}{f+5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(f-5\right)^{2} and f+5 is \left(f+5\right)\left(f-5\right)^{2}. Multiply \frac{f\left(8f^{2}-5\right)}{\left(f-5\right)^{2}} times \frac{f+5}{f+5}. Multiply \frac{9-f}{f+5} times \frac{\left(f-5\right)^{2}}{\left(f-5\right)^{2}}.

$$\frac{f\left(8f^{2}-5\right)\left(f+5\right)}{\left(f+5\right)\left(f-5\right)^{2}}-\frac{\left(9-f\right)\left(f-5\right)^{2}}{\left(f+5\right)\left(f-5\right)^{2}}$$

Since \frac{f\left(8f^{2}-5\right)\left(f+5\right)}{\left(f+5\right)\left(f-5\right)^{2}} and \frac{\left(9-f\right)\left(f-5\right)^{2}}{\left(f+5\right)\left(f-5\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{f\left(8f^{2}-5\right)\left(f+5\right)-\left(9-f\right)\left(f-5\right)^{2}}{\left(f+5\right)\left(f-5\right)^{2}}$$

Do the multiplications in f\left(8f^{2}-5\right)\left(f+5\right)-\left(9-f\right)\left(f-5\right)^{2}.

$$\frac{8f^{4}+40f^{3}-5f^{2}-25f-9f^{2}+90f-225+f^{3}-10f^{2}+25f}{\left(f+5\right)\left(f-5\right)^{2}}$$

Combine like terms in 8f^{4}+40f^{3}-5f^{2}-25f-9f^{2}+90f-225+f^{3}-10f^{2}+25f.

$$\frac{8f^{4}+41f^{3}-24f^{2}+90f-225}{\left(f+5\right)\left(f-5\right)^{2}}$$

Expand \left(f+5\right)\left(f-5\right)^{2}.

$$\frac{8f^{4}+41f^{3}-24f^{2}+90f-225}{f^{3}-5f^{2}-25f+125}$$