Solve (3x^3+5x^2/2-3x+13)(2x^2+2)-(-6x+24)

$$( 3 x ^ { 3 } + 5 x ^ { 2 } / 2 - 3 x + 13 ) ( 2 x ^ { 2 } + 2 ) - ( - 6 x + 24 )$$

Evaluate

$$6x^{5}+5x^{4}+31x^{2}+2$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3x^{3}-3x+13 times \frac{2}{2}.

$$\left(\frac{2\left(3x^{3}-3x+13\right)}{2}+\frac{5x^{2}}{2}\right)\left(2x^{2}+2\right)-\left(-6x+24\right)$$

Since \frac{2\left(3x^{3}-3x+13\right)}{2} and \frac{5x^{2}}{2} have the same denominator, add them by adding their numerators.

$$\frac{2\left(3x^{3}-3x+13\right)+5x^{2}}{2}\left(2x^{2}+2\right)-\left(-6x+24\right)$$

Do the multiplications in 2\left(3x^{3}-3x+13\right)+5x^{2}.

$$\frac{6x^{3}-6x+26+5x^{2}}{2}\left(2x^{2}+2\right)-\left(-6x+24\right)$$

Express \frac{6x^{3}-6x+26+5x^{2}}{2}\left(2x^{2}+2\right) as a single fraction.

$$\frac{\left(6x^{3}-6x+26+5x^{2}\right)\left(2x^{2}+2\right)}{2}-\left(-6x+24\right)$$

Use the distributive property to multiply 6x^{3}-6x+26+5x^{2} by 2x^{2}+2 and combine like terms.

$$\frac{12x^{5}-12x+62x^{2}+52+10x^{4}}{2}-\left(-6x+24\right)$$

To find the opposite of -6x+24, find the opposite of each term.

$$\frac{12x^{5}-12x+62x^{2}+52+10x^{4}}{2}+6x-24$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 6x-24 times \frac{2}{2}.

$$\frac{12x^{5}-12x+62x^{2}+52+10x^{4}}{2}+\frac{2\left(6x-24\right)}{2}$$

Since \frac{12x^{5}-12x+62x^{2}+52+10x^{4}}{2} and \frac{2\left(6x-24\right)}{2} have the same denominator, add them by adding their numerators.

$$\frac{12x^{5}-12x+62x^{2}+52+10x^{4}+2\left(6x-24\right)}{2}$$

Do the multiplications in 12x^{5}-12x+62x^{2}+52+10x^{4}+2\left(6x-24\right).

$$\frac{12x^{5}-12x+62x^{2}+52+10x^{4}+12x-48}{2}$$

Combine like terms in 12x^{5}-12x+62x^{2}+52+10x^{4}+12x-48.

$$\frac{12x^{5}+62x^{2}+4+10x^{4}}{2}$$

Divide each term of 12x^{5}+62x^{2}+4+10x^{4} by 2 to get 6x^{5}+31x^{2}+2+5x^{4}.

$$6x^{5}+31x^{2}+2+5x^{4}$$

Expand

$$6x^{5}+5x^{4}+31x^{2}+2$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3x^{3}-3x+13 times \frac{2}{2}.

$$\left(\frac{2\left(3x^{3}-3x+13\right)}{2}+\frac{5x^{2}}{2}\right)\left(2x^{2}+2\right)-\left(-6x+24\right)$$

Since \frac{2\left(3x^{3}-3x+13\right)}{2} and \frac{5x^{2}}{2} have the same denominator, add them by adding their numerators.

$$\frac{2\left(3x^{3}-3x+13\right)+5x^{2}}{2}\left(2x^{2}+2\right)-\left(-6x+24\right)$$

Do the multiplications in 2\left(3x^{3}-3x+13\right)+5x^{2}.

$$\frac{6x^{3}-6x+26+5x^{2}}{2}\left(2x^{2}+2\right)-\left(-6x+24\right)$$

Express \frac{6x^{3}-6x+26+5x^{2}}{2}\left(2x^{2}+2\right) as a single fraction.

$$\frac{\left(6x^{3}-6x+26+5x^{2}\right)\left(2x^{2}+2\right)}{2}-\left(-6x+24\right)$$

Use the distributive property to multiply 6x^{3}-6x+26+5x^{2} by 2x^{2}+2 and combine like terms.

$$\frac{12x^{5}-12x+62x^{2}+52+10x^{4}}{2}-\left(-6x+24\right)$$

To find the opposite of -6x+24, find the opposite of each term.

$$\frac{12x^{5}-12x+62x^{2}+52+10x^{4}}{2}+6x-24$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 6x-24 times \frac{2}{2}.

$$\frac{12x^{5}-12x+62x^{2}+52+10x^{4}}{2}+\frac{2\left(6x-24\right)}{2}$$

Since \frac{12x^{5}-12x+62x^{2}+52+10x^{4}}{2} and \frac{2\left(6x-24\right)}{2} have the same denominator, add them by adding their numerators.

$$\frac{12x^{5}-12x+62x^{2}+52+10x^{4}+2\left(6x-24\right)}{2}$$

Do the multiplications in 12x^{5}-12x+62x^{2}+52+10x^{4}+2\left(6x-24\right).

$$\frac{12x^{5}-12x+62x^{2}+52+10x^{4}+12x-48}{2}$$

Combine like terms in 12x^{5}-12x+62x^{2}+52+10x^{4}+12x-48.

$$\frac{12x^{5}+62x^{2}+4+10x^{4}}{2}$$

Divide each term of 12x^{5}+62x^{2}+4+10x^{4} by 2 to get 6x^{5}+31x^{2}+2+5x^{4}.

$$6x^{5}+31x^{2}+2+5x^{4}$$