Solve (1/4x+6)(1/3x+12)(2/5x-4)(x+3)

$$( \frac{ 1 }{ 4 } x+6)( \frac{ 1 }{ 3 } x+12)( \frac{ 2 }{ 5 } x-4)(x+3)$$

Evaluate

$$\frac{\left(x-10\right)\left(x+3\right)\left(x+24\right)\left(x+36\right)}{30}$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{1}{4}x+6 by each term of \frac{1}{3}x+12.

$$\left(\frac{1}{4}x\times \frac{1}{3}x+\frac{1}{4}x\times 12+6\times \frac{1}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Multiply x and x to get x^{2}.

$$\left(\frac{1}{4}x^{2}\times \frac{1}{3}+\frac{1}{4}x\times 12+6\times \frac{1}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Multiply \frac{1}{4} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{1\times 1}{4\times 3}x^{2}+\frac{1}{4}x\times 12+6\times \frac{1}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Do the multiplications in the fraction \frac{1\times 1}{4\times 3}.

$$\left(\frac{1}{12}x^{2}+\frac{1}{4}x\times 12+6\times \frac{1}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Multiply \frac{1}{4} and 12 to get \frac{12}{4}.

$$\left(\frac{1}{12}x^{2}+\frac{12}{4}x+6\times \frac{1}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Divide 12 by 4 to get 3.

$$\left(\frac{1}{12}x^{2}+3x+6\times \frac{1}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Multiply 6 and \frac{1}{3} to get \frac{6}{3}.

$$\left(\frac{1}{12}x^{2}+3x+\frac{6}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Divide 6 by 3 to get 2.

$$\left(\frac{1}{12}x^{2}+3x+2x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Combine 3x and 2x to get 5x.

$$\left(\frac{1}{12}x^{2}+5x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Apply the distributive property by multiplying each term of \frac{1}{12}x^{2}+5x+72 by each term of \frac{2}{5}x-4.

$$\left(\frac{1}{12}x^{2}\times \frac{2}{5}x+\frac{1}{12}x^{2}\left(-4\right)+5x\times \frac{2}{5}x-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\left(\frac{1}{12}x^{3}\times \frac{2}{5}+\frac{1}{12}x^{2}\left(-4\right)+5x\times \frac{2}{5}x-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Multiply x and x to get x^{2}.

$$\left(\frac{1}{12}x^{3}\times \frac{2}{5}+\frac{1}{12}x^{2}\left(-4\right)+5x^{2}\times \frac{2}{5}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Multiply \frac{1}{12} times \frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{1\times 2}{12\times 5}x^{3}+\frac{1}{12}x^{2}\left(-4\right)+5x^{2}\times \frac{2}{5}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Do the multiplications in the fraction \frac{1\times 2}{12\times 5}.

$$\left(\frac{2}{60}x^{3}+\frac{1}{12}x^{2}\left(-4\right)+5x^{2}\times \frac{2}{5}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Reduce the fraction \frac{2}{60} to lowest terms by extracting and canceling out 2.

$$\left(\frac{1}{30}x^{3}+\frac{1}{12}x^{2}\left(-4\right)+5x^{2}\times \frac{2}{5}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Multiply \frac{1}{12} and -4 to get \frac{-4}{12}.

$$\left(\frac{1}{30}x^{3}+\frac{-4}{12}x^{2}+5x^{2}\times \frac{2}{5}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Reduce the fraction \frac{-4}{12} to lowest terms by extracting and canceling out 4.

$$\left(\frac{1}{30}x^{3}-\frac{1}{3}x^{2}+5x^{2}\times \frac{2}{5}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Cancel out 5 and 5.

$$\left(\frac{1}{30}x^{3}-\frac{1}{3}x^{2}+2x^{2}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Combine -\frac{1}{3}x^{2} and 2x^{2} to get \frac{5}{3}x^{2}.

$$\left(\frac{1}{30}x^{3}+\frac{5}{3}x^{2}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Express 72\times \frac{2}{5} as a single fraction.

$$\left(\frac{1}{30}x^{3}+\frac{5}{3}x^{2}-20x+\frac{72\times 2}{5}x-288\right)\left(x+3\right)$$

Multiply 72 and 2 to get 144.

$$\left(\frac{1}{30}x^{3}+\frac{5}{3}x^{2}-20x+\frac{144}{5}x-288\right)\left(x+3\right)$$

Combine -20x and \frac{144}{5}x to get \frac{44}{5}x.

$$\left(\frac{1}{30}x^{3}+\frac{5}{3}x^{2}+\frac{44}{5}x-288\right)\left(x+3\right)$$

Apply the distributive property by multiplying each term of \frac{1}{30}x^{3}+\frac{5}{3}x^{2}+\frac{44}{5}x-288 by each term of x+3.

$$\frac{1}{30}x^{3}x+\frac{1}{30}x^{3}\times 3+\frac{5}{3}x^{2}x+\frac{5}{3}x^{2}\times 3+\frac{44}{5}xx+\frac{44}{5}x\times 3-288x-864$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\frac{1}{30}x^{4}+\frac{1}{30}x^{3}\times 3+\frac{5}{3}x^{2}x+\frac{5}{3}x^{2}\times 3+\frac{44}{5}xx+\frac{44}{5}x\times 3-288x-864$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{30}x^{4}+\frac{1}{30}x^{3}\times 3+\frac{5}{3}x^{3}+\frac{5}{3}x^{2}\times 3+\frac{44}{5}xx+\frac{44}{5}x\times 3-288x-864$$

Multiply x and x to get x^{2}.

$$\frac{1}{30}x^{4}+\frac{1}{30}x^{3}\times 3+\frac{5}{3}x^{3}+\frac{5}{3}x^{2}\times 3+\frac{44}{5}x^{2}+\frac{44}{5}x\times 3-288x-864$$

Multiply \frac{1}{30} and 3 to get \frac{3}{30}.

$$\frac{1}{30}x^{4}+\frac{3}{30}x^{3}+\frac{5}{3}x^{3}+\frac{5}{3}x^{2}\times 3+\frac{44}{5}x^{2}+\frac{44}{5}x\times 3-288x-864$$

Reduce the fraction \frac{3}{30} to lowest terms by extracting and canceling out 3.

$$\frac{1}{30}x^{4}+\frac{1}{10}x^{3}+\frac{5}{3}x^{3}+\frac{5}{3}x^{2}\times 3+\frac{44}{5}x^{2}+\frac{44}{5}x\times 3-288x-864$$

Combine \frac{1}{10}x^{3} and \frac{5}{3}x^{3} to get \frac{53}{30}x^{3}.

$$\frac{1}{30}x^{4}+\frac{53}{30}x^{3}+\frac{5}{3}x^{2}\times 3+\frac{44}{5}x^{2}+\frac{44}{5}x\times 3-288x-864$$

Cancel out 3 and 3.

$$\frac{1}{30}x^{4}+\frac{53}{30}x^{3}+5x^{2}+\frac{44}{5}x^{2}+\frac{44}{5}x\times 3-288x-864$$

Combine 5x^{2} and \frac{44}{5}x^{2} to get \frac{69}{5}x^{2}.

$$\frac{1}{30}x^{4}+\frac{53}{30}x^{3}+\frac{69}{5}x^{2}+\frac{44}{5}x\times 3-288x-864$$

Express \frac{44}{5}\times 3 as a single fraction.

$$\frac{1}{30}x^{4}+\frac{53}{30}x^{3}+\frac{69}{5}x^{2}+\frac{44\times 3}{5}x-288x-864$$

Multiply 44 and 3 to get 132.

$$\frac{1}{30}x^{4}+\frac{53}{30}x^{3}+\frac{69}{5}x^{2}+\frac{132}{5}x-288x-864$$

Combine \frac{132}{5}x and -288x to get -\frac{1308}{5}x.

$$\frac{1}{30}x^{4}+\frac{53}{30}x^{3}+\frac{69}{5}x^{2}-\frac{1308}{5}x-864$$

Expand

$$\frac{x^{4}}{30}+\frac{53x^{3}}{30}+\frac{69x^{2}}{5}-\frac{1308x}{5}-864$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{1}{4}x+6 by each term of \frac{1}{3}x+12.

$$\left(\frac{1}{4}x\times \frac{1}{3}x+\frac{1}{4}x\times 12+6\times \frac{1}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Multiply x and x to get x^{2}.

$$\left(\frac{1}{4}x^{2}\times \frac{1}{3}+\frac{1}{4}x\times 12+6\times \frac{1}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Multiply \frac{1}{4} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{1\times 1}{4\times 3}x^{2}+\frac{1}{4}x\times 12+6\times \frac{1}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Do the multiplications in the fraction \frac{1\times 1}{4\times 3}.

$$\left(\frac{1}{12}x^{2}+\frac{1}{4}x\times 12+6\times \frac{1}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Multiply \frac{1}{4} and 12 to get \frac{12}{4}.

$$\left(\frac{1}{12}x^{2}+\frac{12}{4}x+6\times \frac{1}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Divide 12 by 4 to get 3.

$$\left(\frac{1}{12}x^{2}+3x+6\times \frac{1}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Multiply 6 and \frac{1}{3} to get \frac{6}{3}.

$$\left(\frac{1}{12}x^{2}+3x+\frac{6}{3}x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Divide 6 by 3 to get 2.

$$\left(\frac{1}{12}x^{2}+3x+2x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Combine 3x and 2x to get 5x.

$$\left(\frac{1}{12}x^{2}+5x+72\right)\left(\frac{2}{5}x-4\right)\left(x+3\right)$$

Apply the distributive property by multiplying each term of \frac{1}{12}x^{2}+5x+72 by each term of \frac{2}{5}x-4.

$$\left(\frac{1}{12}x^{2}\times \frac{2}{5}x+\frac{1}{12}x^{2}\left(-4\right)+5x\times \frac{2}{5}x-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\left(\frac{1}{12}x^{3}\times \frac{2}{5}+\frac{1}{12}x^{2}\left(-4\right)+5x\times \frac{2}{5}x-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Multiply x and x to get x^{2}.

$$\left(\frac{1}{12}x^{3}\times \frac{2}{5}+\frac{1}{12}x^{2}\left(-4\right)+5x^{2}\times \frac{2}{5}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Multiply \frac{1}{12} times \frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{1\times 2}{12\times 5}x^{3}+\frac{1}{12}x^{2}\left(-4\right)+5x^{2}\times \frac{2}{5}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Do the multiplications in the fraction \frac{1\times 2}{12\times 5}.

$$\left(\frac{2}{60}x^{3}+\frac{1}{12}x^{2}\left(-4\right)+5x^{2}\times \frac{2}{5}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Reduce the fraction \frac{2}{60} to lowest terms by extracting and canceling out 2.

$$\left(\frac{1}{30}x^{3}+\frac{1}{12}x^{2}\left(-4\right)+5x^{2}\times \frac{2}{5}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Multiply \frac{1}{12} and -4 to get \frac{-4}{12}.

$$\left(\frac{1}{30}x^{3}+\frac{-4}{12}x^{2}+5x^{2}\times \frac{2}{5}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Reduce the fraction \frac{-4}{12} to lowest terms by extracting and canceling out 4.

$$\left(\frac{1}{30}x^{3}-\frac{1}{3}x^{2}+5x^{2}\times \frac{2}{5}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Cancel out 5 and 5.

$$\left(\frac{1}{30}x^{3}-\frac{1}{3}x^{2}+2x^{2}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Combine -\frac{1}{3}x^{2} and 2x^{2} to get \frac{5}{3}x^{2}.

$$\left(\frac{1}{30}x^{3}+\frac{5}{3}x^{2}-20x+72\times \frac{2}{5}x-288\right)\left(x+3\right)$$

Express 72\times \frac{2}{5} as a single fraction.

$$\left(\frac{1}{30}x^{3}+\frac{5}{3}x^{2}-20x+\frac{72\times 2}{5}x-288\right)\left(x+3\right)$$

Multiply 72 and 2 to get 144.

$$\left(\frac{1}{30}x^{3}+\frac{5}{3}x^{2}-20x+\frac{144}{5}x-288\right)\left(x+3\right)$$

Combine -20x and \frac{144}{5}x to get \frac{44}{5}x.

$$\left(\frac{1}{30}x^{3}+\frac{5}{3}x^{2}+\frac{44}{5}x-288\right)\left(x+3\right)$$

Apply the distributive property by multiplying each term of \frac{1}{30}x^{3}+\frac{5}{3}x^{2}+\frac{44}{5}x-288 by each term of x+3.

$$\frac{1}{30}x^{3}x+\frac{1}{30}x^{3}\times 3+\frac{5}{3}x^{2}x+\frac{5}{3}x^{2}\times 3+\frac{44}{5}xx+\frac{44}{5}x\times 3-288x-864$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\frac{1}{30}x^{4}+\frac{1}{30}x^{3}\times 3+\frac{5}{3}x^{2}x+\frac{5}{3}x^{2}\times 3+\frac{44}{5}xx+\frac{44}{5}x\times 3-288x-864$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{30}x^{4}+\frac{1}{30}x^{3}\times 3+\frac{5}{3}x^{3}+\frac{5}{3}x^{2}\times 3+\frac{44}{5}xx+\frac{44}{5}x\times 3-288x-864$$

Multiply x and x to get x^{2}.

$$\frac{1}{30}x^{4}+\frac{1}{30}x^{3}\times 3+\frac{5}{3}x^{3}+\frac{5}{3}x^{2}\times 3+\frac{44}{5}x^{2}+\frac{44}{5}x\times 3-288x-864$$

Multiply \frac{1}{30} and 3 to get \frac{3}{30}.

$$\frac{1}{30}x^{4}+\frac{3}{30}x^{3}+\frac{5}{3}x^{3}+\frac{5}{3}x^{2}\times 3+\frac{44}{5}x^{2}+\frac{44}{5}x\times 3-288x-864$$

Reduce the fraction \frac{3}{30} to lowest terms by extracting and canceling out 3.

$$\frac{1}{30}x^{4}+\frac{1}{10}x^{3}+\frac{5}{3}x^{3}+\frac{5}{3}x^{2}\times 3+\frac{44}{5}x^{2}+\frac{44}{5}x\times 3-288x-864$$

Combine \frac{1}{10}x^{3} and \frac{5}{3}x^{3} to get \frac{53}{30}x^{3}.

$$\frac{1}{30}x^{4}+\frac{53}{30}x^{3}+\frac{5}{3}x^{2}\times 3+\frac{44}{5}x^{2}+\frac{44}{5}x\times 3-288x-864$$

Cancel out 3 and 3.

$$\frac{1}{30}x^{4}+\frac{53}{30}x^{3}+5x^{2}+\frac{44}{5}x^{2}+\frac{44}{5}x\times 3-288x-864$$

Combine 5x^{2} and \frac{44}{5}x^{2} to get \frac{69}{5}x^{2}.

$$\frac{1}{30}x^{4}+\frac{53}{30}x^{3}+\frac{69}{5}x^{2}+\frac{44}{5}x\times 3-288x-864$$

Express \frac{44}{5}\times 3 as a single fraction.

$$\frac{1}{30}x^{4}+\frac{53}{30}x^{3}+\frac{69}{5}x^{2}+\frac{44\times 3}{5}x-288x-864$$

Multiply 44 and 3 to get 132.

$$\frac{1}{30}x^{4}+\frac{53}{30}x^{3}+\frac{69}{5}x^{2}+\frac{132}{5}x-288x-864$$

Combine \frac{132}{5}x and -288x to get -\frac{1308}{5}x.

$$\frac{1}{30}x^{4}+\frac{53}{30}x^{3}+\frac{69}{5}x^{2}-\frac{1308}{5}x-864$$