Solve 1/4(x-2)(2x+3)-4(2x+1)(2x-1)

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

Evaluate

$$-\frac{31x^{2}}{2}-\frac{x}{4}+\frac{5}{2}$$

Solution Steps

Use the distributive property to multiply \frac{1}{4} by x-2.

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

Multiply \frac{1}{4} and -2 to get \frac{-2}{4}.

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

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

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

Apply the distributive property by multiplying each term of \frac{1}{4}x-\frac{1}{2} by each term of 2x+3.

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

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

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

Multiply \frac{1}{4} and 2 to get \frac{2}{4}.

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

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

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

Multiply \frac{1}{4} and 3 to get \frac{3}{4}.

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

Cancel out 2 and 2.

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

Combine \frac{3}{4}x and -x to get -\frac{1}{4}x.

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

Express -\frac{1}{2}\times 3 as a single fraction.

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

Fraction \frac{-3}{2} can be rewritten as -\frac{3}{2} by extracting the negative sign.

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

Use the distributive property to multiply -4 by 2x+1.

$$\frac{1}{2}x^{2}-\frac{1}{4}x-\frac{3}{2}+\left(-8x-4\right)\left(2x-1\right)$$

Apply the distributive property by multiplying each term of -8x-4 by each term of 2x-1.

$$\frac{1}{2}x^{2}-\frac{1}{4}x-\frac{3}{2}-16x^{2}+8x-8x+4$$

Combine 8x and -8x to get 0.

$$\frac{1}{2}x^{2}-\frac{1}{4}x-\frac{3}{2}-16x^{2}+4$$

Combine \frac{1}{2}x^{2} and -16x^{2} to get -\frac{31}{2}x^{2}.

$$-\frac{31}{2}x^{2}-\frac{1}{4}x-\frac{3}{2}+4$$

Convert 4 to fraction \frac{8}{2}.

$$-\frac{31}{2}x^{2}-\frac{1}{4}x-\frac{3}{2}+\frac{8}{2}$$

Since -\frac{3}{2} and \frac{8}{2} have the same denominator, add them by adding their numerators.

$$-\frac{31}{2}x^{2}-\frac{1}{4}x+\frac{-3+8}{2}$$

Add -3 and 8 to get 5.

$$-\frac{31}{2}x^{2}-\frac{1}{4}x+\frac{5}{2}$$

Expand

$$-\frac{31x^{2}}{2}-\frac{x}{4}+\frac{5}{2}$$

Solution Steps

Use the distributive property to multiply \frac{1}{4} by x-2.

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

Multiply \frac{1}{4} and -2 to get \frac{-2}{4}.

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

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

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

Apply the distributive property by multiplying each term of \frac{1}{4}x-\frac{1}{2} by each term of 2x+3.

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

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

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

Multiply \frac{1}{4} and 2 to get \frac{2}{4}.

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

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

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

Multiply \frac{1}{4} and 3 to get \frac{3}{4}.

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

Cancel out 2 and 2.

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

Combine \frac{3}{4}x and -x to get -\frac{1}{4}x.

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

Express -\frac{1}{2}\times 3 as a single fraction.

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

Fraction \frac{-3}{2} can be rewritten as -\frac{3}{2} by extracting the negative sign.

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

Use the distributive property to multiply -4 by 2x+1.

$$\frac{1}{2}x^{2}-\frac{1}{4}x-\frac{3}{2}+\left(-8x-4\right)\left(2x-1\right)$$

Apply the distributive property by multiplying each term of -8x-4 by each term of 2x-1.

$$\frac{1}{2}x^{2}-\frac{1}{4}x-\frac{3}{2}-16x^{2}+8x-8x+4$$

Combine 8x and -8x to get 0.

$$\frac{1}{2}x^{2}-\frac{1}{4}x-\frac{3}{2}-16x^{2}+4$$

Combine \frac{1}{2}x^{2} and -16x^{2} to get -\frac{31}{2}x^{2}.

$$-\frac{31}{2}x^{2}-\frac{1}{4}x-\frac{3}{2}+4$$

Convert 4 to fraction \frac{8}{2}.

$$-\frac{31}{2}x^{2}-\frac{1}{4}x-\frac{3}{2}+\frac{8}{2}$$

Since -\frac{3}{2} and \frac{8}{2} have the same denominator, add them by adding their numerators.

$$-\frac{31}{2}x^{2}-\frac{1}{4}x+\frac{-3+8}{2}$$

Add -3 and 8 to get 5.

$$-\frac{31}{2}x^{2}-\frac{1}{4}x+\frac{5}{2}$$