Solve left(x+5right)^3-left(x+1right)^3-4(3x^2-5)-10x-7=

$${ \left(x+5 \right) }^{ 3 } - { \left(x+1 \right) }^{ 3 } -4(3 { x }^{ 2 } -5)-10x-7=$$

Evaluate

$$62x+137$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+5\right)^{3}.

$$x^{3}+15x^{2}+75x+125-\left(x+1\right)^{3}-4\left(3x^{2}-5\right)-10x-7$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+1\right)^{3}.

$$x^{3}+15x^{2}+75x+125-\left(x^{3}+3x^{2}+3x+1\right)-4\left(3x^{2}-5\right)-10x-7$$

To find the opposite of x^{3}+3x^{2}+3x+1, find the opposite of each term.

$$x^{3}+15x^{2}+75x+125-x^{3}-3x^{2}-3x-1-4\left(3x^{2}-5\right)-10x-7$$

Combine x^{3} and -x^{3} to get 0.

$$15x^{2}+75x+125-3x^{2}-3x-1-4\left(3x^{2}-5\right)-10x-7$$

Combine 15x^{2} and -3x^{2} to get 12x^{2}.

$$12x^{2}+75x+125-3x-1-4\left(3x^{2}-5\right)-10x-7$$

Combine 75x and -3x to get 72x.

$$12x^{2}+72x+125-1-4\left(3x^{2}-5\right)-10x-7$$

Subtract 1 from 125 to get 124.

$$12x^{2}+72x+124-4\left(3x^{2}-5\right)-10x-7$$

Use the distributive property to multiply -4 by 3x^{2}-5.

$$12x^{2}+72x+124-12x^{2}+20-10x-7$$

Combine 12x^{2} and -12x^{2} to get 0.

$$72x+124+20-10x-7$$

Add 124 and 20 to get 144.

$$72x+144-10x-7$$

Combine 72x and -10x to get 62x.

$$62x+144-7$$

Subtract 7 from 144 to get 137.

$$62x+137$$

Expand

$$62x+137$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+5\right)^{3}.

$$x^{3}+15x^{2}+75x+125-\left(x+1\right)^{3}-4\left(3x^{2}-5\right)-10x-7$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+1\right)^{3}.

$$x^{3}+15x^{2}+75x+125-\left(x^{3}+3x^{2}+3x+1\right)-4\left(3x^{2}-5\right)-10x-7$$

To find the opposite of x^{3}+3x^{2}+3x+1, find the opposite of each term.

$$x^{3}+15x^{2}+75x+125-x^{3}-3x^{2}-3x-1-4\left(3x^{2}-5\right)-10x-7$$

Combine x^{3} and -x^{3} to get 0.

$$15x^{2}+75x+125-3x^{2}-3x-1-4\left(3x^{2}-5\right)-10x-7$$

Combine 15x^{2} and -3x^{2} to get 12x^{2}.

$$12x^{2}+75x+125-3x-1-4\left(3x^{2}-5\right)-10x-7$$

Combine 75x and -3x to get 72x.

$$12x^{2}+72x+125-1-4\left(3x^{2}-5\right)-10x-7$$

Subtract 1 from 125 to get 124.

$$12x^{2}+72x+124-4\left(3x^{2}-5\right)-10x-7$$

Use the distributive property to multiply -4 by 3x^{2}-5.

$$12x^{2}+72x+124-12x^{2}+20-10x-7$$

Combine 12x^{2} and -12x^{2} to get 0.

$$72x+124+20-10x-7$$

Add 124 and 20 to get 144.

$$72x+144-10x-7$$

Combine 72x and -10x to get 62x.

$$62x+144-7$$

Subtract 7 from 144 to get 137.

$$62x+137$$