Solve (4x^2-3x+7/2)*(-3x^3+6x^2-3/4x+3/5)

$$(4 { x }^{ 2 } -3x+ \frac{ 7 }{ 2 } ) \times (-3 { x }^{ 3 } +6 { x }^{ 2 } - \frac{ 3 }{ 4x } + \frac{ 3 }{ 5 } )$$

Evaluate

$$-\frac{3\left(8x^{2}-6x+7\right)\left(20x^{4}-40x^{3}-4x+5\right)}{40x}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4x and 5 is 20x. Multiply \frac{3}{4x} times \frac{5}{5}. Multiply \frac{3}{5} times \frac{4x}{4x}.

$$\left(4x^{2}-3x+\frac{7}{2}\right)\left(-3x^{3}+6x^{2}-\frac{3\times 5}{20x}+\frac{3\times 4x}{20x}\right)$$

Since -\frac{3\times 5}{20x} and \frac{3\times 4x}{20x} have the same denominator, add them by adding their numerators.

$$\left(4x^{2}-3x+\frac{7}{2}\right)\left(-3x^{3}+6x^{2}+\frac{-3\times 5+3\times 4x}{20x}\right)$$

Do the multiplications in -3\times 5+3\times 4x.

$$\left(4x^{2}-3x+\frac{7}{2}\right)\left(-3x^{3}+6x^{2}+\frac{-15+12x}{20x}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -3x^{3}+6x^{2} times \frac{20x}{20x}.

$$\left(4x^{2}-3x+\frac{7}{2}\right)\left(\frac{\left(-3x^{3}+6x^{2}\right)\times 20x}{20x}+\frac{-15+12x}{20x}\right)$$

Since \frac{\left(-3x^{3}+6x^{2}\right)\times 20x}{20x} and \frac{-15+12x}{20x} have the same denominator, add them by adding their numerators.

$$\left(4x^{2}-3x+\frac{7}{2}\right)\times \frac{\left(-3x^{3}+6x^{2}\right)\times 20x-15+12x}{20x}$$

Do the multiplications in \left(-3x^{3}+6x^{2}\right)\times 20x-15+12x.

$$\left(4x^{2}-3x+\frac{7}{2}\right)\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}$$

Use the distributive property to multiply 4x^{2}-3x+\frac{7}{2} by \frac{-60x^{4}+120x^{3}-15+12x}{20x}.

$$4x^{2}\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}-3x\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}+\frac{7}{2}\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}$$

Express 4\times \frac{-60x^{4}+120x^{3}-15+12x}{20x} as a single fraction.

$$\frac{4\left(-60x^{4}+120x^{3}-15+12x\right)}{20x}x^{2}-3x\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}+\frac{7}{2}\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}$$

Cancel out 4 in both numerator and denominator.

$$\frac{-60x^{4}+120x^{3}+12x-15}{5x}x^{2}-3x\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}+\frac{7}{2}\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}$$

Express -3\times \frac{-60x^{4}+120x^{3}-15+12x}{20x} as a single fraction.

$$\frac{-60x^{4}+120x^{3}+12x-15}{5x}x^{2}+\frac{-3\left(-60x^{4}+120x^{3}-15+12x\right)}{20x}x+\frac{7}{2}\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}$$

Multiply \frac{7}{2} times \frac{-60x^{4}+120x^{3}-15+12x}{20x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-60x^{4}+120x^{3}+12x-15}{5x}x^{2}+\frac{-3\left(-60x^{4}+120x^{3}-15+12x\right)}{20x}x+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{2\times 20x}$$

Express \frac{-60x^{4}+120x^{3}+12x-15}{5x}x^{2} as a single fraction.

$$\frac{\left(-60x^{4}+120x^{3}+12x-15\right)x^{2}}{5x}+\frac{-3\left(-60x^{4}+120x^{3}-15+12x\right)}{20x}x+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{2\times 20x}$$

Cancel out x in both numerator and denominator.

$$\frac{x\left(-60x^{4}+120x^{3}+12x-15\right)}{5}+\frac{-3\left(-60x^{4}+120x^{3}-15+12x\right)}{20x}x+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{2\times 20x}$$

Express \frac{-3\left(-60x^{4}+120x^{3}-15+12x\right)}{20x}x as a single fraction.

$$\frac{x\left(-60x^{4}+120x^{3}+12x-15\right)}{5}+\frac{-3\left(-60x^{4}+120x^{3}-15+12x\right)x}{20x}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{2\times 20x}$$

Cancel out x in both numerator and denominator.

$$\frac{x\left(-60x^{4}+120x^{3}+12x-15\right)}{5}+\frac{-3\left(-60x^{4}+120x^{3}+12x-15\right)}{20}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{2\times 20x}$$

Multiply 2 and 20 to get 40.

$$\frac{x\left(-60x^{4}+120x^{3}+12x-15\right)}{5}+\frac{-3\left(-60x^{4}+120x^{3}+12x-15\right)}{20}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 20 is 20. Multiply \frac{x\left(-60x^{4}+120x^{3}+12x-15\right)}{5} times \frac{4}{4}.

$$\frac{4x\left(-60x^{4}+120x^{3}+12x-15\right)}{20}+\frac{-3\left(-60x^{4}+120x^{3}+12x-15\right)}{20}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

Since \frac{4x\left(-60x^{4}+120x^{3}+12x-15\right)}{20} and \frac{-3\left(-60x^{4}+120x^{3}+12x-15\right)}{20} have the same denominator, add them by adding their numerators.

$$\frac{4x\left(-60x^{4}+120x^{3}+12x-15\right)-3\left(-60x^{4}+120x^{3}+12x-15\right)}{20}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

Do the multiplications in 4x\left(-60x^{4}+120x^{3}+12x-15\right)-3\left(-60x^{4}+120x^{3}+12x-15\right).

$$\frac{-240x^{5}+480x^{4}+48x^{2}-60x+180x^{4}-360x^{3}-36x+45}{20}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

Combine like terms in -240x^{5}+480x^{4}+48x^{2}-60x+180x^{4}-360x^{3}-36x+45.

$$\frac{-240x^{5}+660x^{4}+48x^{2}-96x-360x^{3}+45}{20}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 20 and 40x is 40x. Multiply \frac{-240x^{5}+660x^{4}+48x^{2}-96x-360x^{3}+45}{20} times \frac{2x}{2x}.

$$\frac{\left(-240x^{5}+660x^{4}+48x^{2}-96x-360x^{3}+45\right)\times 2x}{40x}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

Since \frac{\left(-240x^{5}+660x^{4}+48x^{2}-96x-360x^{3}+45\right)\times 2x}{40x} and \frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x} have the same denominator, add them by adding their numerators.

$$\frac{\left(-240x^{5}+660x^{4}+48x^{2}-96x-360x^{3}+45\right)\times 2x+7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

Do the multiplications in \left(-240x^{5}+660x^{4}+48x^{2}-96x-360x^{3}+45\right)\times 2x+7\left(-60x^{4}+120x^{3}-15+12x\right).

$$\frac{-480x^{6}+1320x^{5}+96x^{3}-192x^{2}-720x^{4}+90x-420x^{4}+840x^{3}-105+84x}{40x}$$

Combine like terms in -480x^{6}+1320x^{5}+96x^{3}-192x^{2}-720x^{4}+90x-420x^{4}+840x^{3}-105+84x.

$$\frac{174x-480x^{6}+1320x^{5}+936x^{3}-192x^{2}-1140x^{4}-105}{40x}$$

Expand

$$-12x^{5}+33x^{4}-\frac{57x^{3}}{2}+\frac{117x^{2}}{5}-\frac{24x}{5}+\frac{87}{20}-\frac{21}{8x}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4x and 5 is 20x. Multiply \frac{3}{4x} times \frac{5}{5}. Multiply \frac{3}{5} times \frac{4x}{4x}.

$$\left(4x^{2}-3x+\frac{7}{2}\right)\left(-3x^{3}+6x^{2}-\frac{3\times 5}{20x}+\frac{3\times 4x}{20x}\right)$$

Since -\frac{3\times 5}{20x} and \frac{3\times 4x}{20x} have the same denominator, add them by adding their numerators.

$$\left(4x^{2}-3x+\frac{7}{2}\right)\left(-3x^{3}+6x^{2}+\frac{-3\times 5+3\times 4x}{20x}\right)$$

Do the multiplications in -3\times 5+3\times 4x.

$$\left(4x^{2}-3x+\frac{7}{2}\right)\left(-3x^{3}+6x^{2}+\frac{-15+12x}{20x}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -3x^{3}+6x^{2} times \frac{20x}{20x}.

$$\left(4x^{2}-3x+\frac{7}{2}\right)\left(\frac{\left(-3x^{3}+6x^{2}\right)\times 20x}{20x}+\frac{-15+12x}{20x}\right)$$

Since \frac{\left(-3x^{3}+6x^{2}\right)\times 20x}{20x} and \frac{-15+12x}{20x} have the same denominator, add them by adding their numerators.

$$\left(4x^{2}-3x+\frac{7}{2}\right)\times \frac{\left(-3x^{3}+6x^{2}\right)\times 20x-15+12x}{20x}$$

Do the multiplications in \left(-3x^{3}+6x^{2}\right)\times 20x-15+12x.

$$\left(4x^{2}-3x+\frac{7}{2}\right)\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}$$

Use the distributive property to multiply 4x^{2}-3x+\frac{7}{2} by \frac{-60x^{4}+120x^{3}-15+12x}{20x}.

$$4x^{2}\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}-3x\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}+\frac{7}{2}\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}$$

Express 4\times \frac{-60x^{4}+120x^{3}-15+12x}{20x} as a single fraction.

$$\frac{4\left(-60x^{4}+120x^{3}-15+12x\right)}{20x}x^{2}-3x\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}+\frac{7}{2}\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}$$

Cancel out 4 in both numerator and denominator.

$$\frac{-60x^{4}+120x^{3}+12x-15}{5x}x^{2}-3x\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}+\frac{7}{2}\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}$$

Express -3\times \frac{-60x^{4}+120x^{3}-15+12x}{20x} as a single fraction.

$$\frac{-60x^{4}+120x^{3}+12x-15}{5x}x^{2}+\frac{-3\left(-60x^{4}+120x^{3}-15+12x\right)}{20x}x+\frac{7}{2}\times \frac{-60x^{4}+120x^{3}-15+12x}{20x}$$

Multiply \frac{7}{2} times \frac{-60x^{4}+120x^{3}-15+12x}{20x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-60x^{4}+120x^{3}+12x-15}{5x}x^{2}+\frac{-3\left(-60x^{4}+120x^{3}-15+12x\right)}{20x}x+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{2\times 20x}$$

Express \frac{-60x^{4}+120x^{3}+12x-15}{5x}x^{2} as a single fraction.

$$\frac{\left(-60x^{4}+120x^{3}+12x-15\right)x^{2}}{5x}+\frac{-3\left(-60x^{4}+120x^{3}-15+12x\right)}{20x}x+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{2\times 20x}$$

Cancel out x in both numerator and denominator.

$$\frac{x\left(-60x^{4}+120x^{3}+12x-15\right)}{5}+\frac{-3\left(-60x^{4}+120x^{3}-15+12x\right)}{20x}x+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{2\times 20x}$$

Express \frac{-3\left(-60x^{4}+120x^{3}-15+12x\right)}{20x}x as a single fraction.

$$\frac{x\left(-60x^{4}+120x^{3}+12x-15\right)}{5}+\frac{-3\left(-60x^{4}+120x^{3}-15+12x\right)x}{20x}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{2\times 20x}$$

Cancel out x in both numerator and denominator.

$$\frac{x\left(-60x^{4}+120x^{3}+12x-15\right)}{5}+\frac{-3\left(-60x^{4}+120x^{3}+12x-15\right)}{20}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{2\times 20x}$$

Multiply 2 and 20 to get 40.

$$\frac{x\left(-60x^{4}+120x^{3}+12x-15\right)}{5}+\frac{-3\left(-60x^{4}+120x^{3}+12x-15\right)}{20}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 20 is 20. Multiply \frac{x\left(-60x^{4}+120x^{3}+12x-15\right)}{5} times \frac{4}{4}.

$$\frac{4x\left(-60x^{4}+120x^{3}+12x-15\right)}{20}+\frac{-3\left(-60x^{4}+120x^{3}+12x-15\right)}{20}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

Since \frac{4x\left(-60x^{4}+120x^{3}+12x-15\right)}{20} and \frac{-3\left(-60x^{4}+120x^{3}+12x-15\right)}{20} have the same denominator, add them by adding their numerators.

$$\frac{4x\left(-60x^{4}+120x^{3}+12x-15\right)-3\left(-60x^{4}+120x^{3}+12x-15\right)}{20}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

Do the multiplications in 4x\left(-60x^{4}+120x^{3}+12x-15\right)-3\left(-60x^{4}+120x^{3}+12x-15\right).

$$\frac{-240x^{5}+480x^{4}+48x^{2}-60x+180x^{4}-360x^{3}-36x+45}{20}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

Combine like terms in -240x^{5}+480x^{4}+48x^{2}-60x+180x^{4}-360x^{3}-36x+45.

$$\frac{-240x^{5}+660x^{4}+48x^{2}-96x-360x^{3}+45}{20}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 20 and 40x is 40x. Multiply \frac{-240x^{5}+660x^{4}+48x^{2}-96x-360x^{3}+45}{20} times \frac{2x}{2x}.

$$\frac{\left(-240x^{5}+660x^{4}+48x^{2}-96x-360x^{3}+45\right)\times 2x}{40x}+\frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

Since \frac{\left(-240x^{5}+660x^{4}+48x^{2}-96x-360x^{3}+45\right)\times 2x}{40x} and \frac{7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x} have the same denominator, add them by adding their numerators.

$$\frac{\left(-240x^{5}+660x^{4}+48x^{2}-96x-360x^{3}+45\right)\times 2x+7\left(-60x^{4}+120x^{3}-15+12x\right)}{40x}$$

Do the multiplications in \left(-240x^{5}+660x^{4}+48x^{2}-96x-360x^{3}+45\right)\times 2x+7\left(-60x^{4}+120x^{3}-15+12x\right).

$$\frac{-480x^{6}+1320x^{5}+96x^{3}-192x^{2}-720x^{4}+90x-420x^{4}+840x^{3}-105+84x}{40x}$$

Combine like terms in -480x^{6}+1320x^{5}+96x^{3}-192x^{2}-720x^{4}+90x-420x^{4}+840x^{3}-105+84x.

$$\frac{174x-480x^{6}+1320x^{5}+936x^{3}-192x^{2}-1140x^{4}-105}{40x}$$