Solve e+2f/5e+2f*4f^2-10ef/3e^2-9ef

$$\frac { e + 2 f } { 5 e + 2 f } \times \frac { 4 f ^ { 2 } - 10 e f } { 3 e ^ { 2 } - 9 e f }$$

Evaluate

$$\frac{2f\left(2f-5e\right)\left(2f+e\right)}{3e\left(e-3f\right)\left(2f+5e\right)}$$

Short Solution Steps

Multiply \frac{e+2f}{5e+2f} times \frac{4f^{2}-10ef}{3e^{2}-9ef} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(e+2f\right)\left(4f^{2}-10ef\right)}{\left(5e+2f\right)\left(3e^{2}-9ef\right)}$$

Use the distributive property to multiply e+2f by 4f^{2}-10ef.

$$\frac{e\left(4f^{2}-10ef\right)+2f\left(4f^{2}-10ef\right)}{\left(5e+2f\right)\left(3e^{2}-9ef\right)}$$

Use the distributive property to multiply 5e+2f by 3e^{2}-9ef.

$$\frac{e\left(4f^{2}-10ef\right)+2f\left(4f^{2}-10ef\right)}{5e\left(3e^{2}-9ef\right)+2f\left(3e^{2}-9ef\right)}$$

Use the distributive property to multiply e by 4f^{2}-10ef.

$$\frac{4ef^{2}-10fe^{2}+2f\left(4f^{2}-10ef\right)}{5e\left(3e^{2}-9ef\right)+2f\left(3e^{2}-9ef\right)}$$

Use the distributive property to multiply 2f by 4f^{2}-10ef.

$$\frac{4ef^{2}-10fe^{2}+8f^{3}-20ef^{2}}{5e\left(3e^{2}-9ef\right)+2f\left(3e^{2}-9ef\right)}$$

Combine 4ef^{2} and -20ef^{2} to get -16ef^{2}.

$$\frac{-16ef^{2}-10fe^{2}+8f^{3}}{5e\left(3e^{2}-9ef\right)+2f\left(3e^{2}-9ef\right)}$$

Use the distributive property to multiply 5e by 3e^{2}-9ef.

$$\frac{-16ef^{2}-10fe^{2}+8f^{3}}{15e^{3}-45e^{2}f+2f\left(3e^{2}-9ef\right)}$$

Use the distributive property to multiply 2f by 3e^{2}-9ef.

$$\frac{-16ef^{2}-10fe^{2}+8f^{3}}{15e^{3}-45e^{2}f+6e^{2}f-18ef^{2}}$$

Combine -45e^{2}f and 6e^{2}f to get -39e^{2}f.

$$\frac{-16ef^{2}-10fe^{2}+8f^{3}}{15e^{3}-39e^{2}f-18ef^{2}}$$

Expand

$$-\frac{2\left(5e^{2}f+8ef^{2}-4f^{3}\right)}{3\left(2f+5e\right)\left(e^{2}-3ef\right)}$$

Short Solution Steps

Multiply \frac{e+2f}{5e+2f} times \frac{4f^{2}-10ef}{3e^{2}-9ef} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(e+2f\right)\left(4f^{2}-10ef\right)}{\left(5e+2f\right)\left(3e^{2}-9ef\right)}$$

Use the distributive property to multiply e+2f by 4f^{2}-10ef.

$$\frac{e\left(4f^{2}-10ef\right)+2f\left(4f^{2}-10ef\right)}{\left(5e+2f\right)\left(3e^{2}-9ef\right)}$$

Use the distributive property to multiply 5e+2f by 3e^{2}-9ef.

$$\frac{e\left(4f^{2}-10ef\right)+2f\left(4f^{2}-10ef\right)}{5e\left(3e^{2}-9ef\right)+2f\left(3e^{2}-9ef\right)}$$

Use the distributive property to multiply e by 4f^{2}-10ef.

$$\frac{4ef^{2}-10fe^{2}+2f\left(4f^{2}-10ef\right)}{5e\left(3e^{2}-9ef\right)+2f\left(3e^{2}-9ef\right)}$$

Use the distributive property to multiply 2f by 4f^{2}-10ef.

$$\frac{4ef^{2}-10fe^{2}+8f^{3}-20ef^{2}}{5e\left(3e^{2}-9ef\right)+2f\left(3e^{2}-9ef\right)}$$

Combine 4ef^{2} and -20ef^{2} to get -16ef^{2}.

$$\frac{-16ef^{2}-10fe^{2}+8f^{3}}{5e\left(3e^{2}-9ef\right)+2f\left(3e^{2}-9ef\right)}$$

Use the distributive property to multiply 5e by 3e^{2}-9ef.

$$\frac{-16ef^{2}-10fe^{2}+8f^{3}}{15e^{3}-45e^{2}f+2f\left(3e^{2}-9ef\right)}$$

Use the distributive property to multiply 2f by 3e^{2}-9ef.

$$\frac{-16ef^{2}-10fe^{2}+8f^{3}}{15e^{3}-45e^{2}f+6e^{2}f-18ef^{2}}$$

Combine -45e^{2}f and 6e^{2}f to get -39e^{2}f.

$$\frac{-16ef^{2}-10fe^{2}+8f^{3}}{15e^{3}-39e^{2}f-18ef^{2}}$$