Solve left(2+xright)left(-x+2right)left(4+x^2right)+left(3x-4right)left(3x+4right)-left(2x^2+5right)left(-5+2x^2right)

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

Evaluate

$$25+9x^{2}-5x^{4}$$

Solution Steps

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

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

Use the distributive property to multiply 2\left(-x\right)+4+x\left(-x\right)+2x by 4+x^{2}.

$$8\left(-x\right)+2\left(-x\right)x^{2}+16+4x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}+\left(3x-4\right)\left(3x+4\right)-\left(2x^{2}+5\right)\left(-5+2x^{2}\right)$$

Consider \left(3x-4\right)\left(3x+4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 4.

$$8\left(-x\right)+2\left(-x\right)x^{2}+16+4x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}+\left(3x\right)^{2}-16-\left(2x^{2}+5\right)\left(-5+2x^{2}\right)$$

Expand \left(3x\right)^{2}.

$$8\left(-x\right)+2\left(-x\right)x^{2}+16+4x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}+3^{2}x^{2}-16-\left(2x^{2}+5\right)\left(-5+2x^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$8\left(-x\right)+2\left(-x\right)x^{2}+16+4x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}+9x^{2}-16-\left(2x^{2}+5\right)\left(-5+2x^{2}\right)$$

Combine 4x^{2} and 9x^{2} to get 13x^{2}.

$$8\left(-x\right)+2\left(-x\right)x^{2}+16+13x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}-16-\left(2x^{2}+5\right)\left(-5+2x^{2}\right)$$

Subtract 16 from 16 to get 0.

$$8\left(-x\right)+2\left(-x\right)x^{2}+13x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}-\left(2x^{2}+5\right)\left(-5+2x^{2}\right)$$

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

$$8\left(-x\right)+2\left(-x\right)x^{2}+13x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}-\left(4x^{4}-25\right)$$

To find the opposite of 4x^{4}-25, find the opposite of each term.

$$8\left(-x\right)+2\left(-x\right)x^{2}+13x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}-4x^{4}+25$$

Multiply 8 and -1 to get -8.

$$-8x+2\left(-1\right)xx^{2}+13x^{2}+4x\left(-1\right)x-xx^{3}+8x+2x^{3}-4x^{4}+25$$

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

$$-8x+2\left(-1\right)x^{3}+13x^{2}+4x\left(-1\right)x-xx^{3}+8x+2x^{3}-4x^{4}+25$$

Multiply 2 and -1 to get -2.

$$-8x-2x^{3}+13x^{2}+4x\left(-1\right)x-xx^{3}+8x+2x^{3}-4x^{4}+25$$

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

$$-8x-2x^{3}+13x^{2}+4x^{2}\left(-1\right)-xx^{3}+8x+2x^{3}-4x^{4}+25$$

Multiply 4 and -1 to get -4.

$$-8x-2x^{3}+13x^{2}-4x^{2}-xx^{3}+8x+2x^{3}-4x^{4}+25$$

Combine 13x^{2} and -4x^{2} to get 9x^{2}.

$$-8x-2x^{3}+9x^{2}-xx^{3}+8x+2x^{3}-4x^{4}+25$$

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

$$-8x-2x^{3}+9x^{2}-x^{4}+8x+2x^{3}-4x^{4}+25$$

Combine -8x and 8x to get 0.

$$-2x^{3}+9x^{2}-x^{4}+2x^{3}-4x^{4}+25$$

Combine -2x^{3} and 2x^{3} to get 0.

$$9x^{2}-x^{4}-4x^{4}+25$$

Combine -x^{4} and -4x^{4} to get -5x^{4}.

$$9x^{2}-5x^{4}+25$$

Expand

$$25+9x^{2}-5x^{4}$$

Solution Steps

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

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

Use the distributive property to multiply 2\left(-x\right)+4+x\left(-x\right)+2x by 4+x^{2}.

$$8\left(-x\right)+2\left(-x\right)x^{2}+16+4x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}+\left(3x-4\right)\left(3x+4\right)-\left(2x^{2}+5\right)\left(-5+2x^{2}\right)$$

Consider \left(3x-4\right)\left(3x+4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 4.

$$8\left(-x\right)+2\left(-x\right)x^{2}+16+4x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}+\left(3x\right)^{2}-16-\left(2x^{2}+5\right)\left(-5+2x^{2}\right)$$

Expand \left(3x\right)^{2}.

$$8\left(-x\right)+2\left(-x\right)x^{2}+16+4x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}+3^{2}x^{2}-16-\left(2x^{2}+5\right)\left(-5+2x^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$8\left(-x\right)+2\left(-x\right)x^{2}+16+4x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}+9x^{2}-16-\left(2x^{2}+5\right)\left(-5+2x^{2}\right)$$

Combine 4x^{2} and 9x^{2} to get 13x^{2}.

$$8\left(-x\right)+2\left(-x\right)x^{2}+16+13x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}-16-\left(2x^{2}+5\right)\left(-5+2x^{2}\right)$$

Subtract 16 from 16 to get 0.

$$8\left(-x\right)+2\left(-x\right)x^{2}+13x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}-\left(2x^{2}+5\right)\left(-5+2x^{2}\right)$$

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

$$8\left(-x\right)+2\left(-x\right)x^{2}+13x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}-\left(4x^{4}-25\right)$$

To find the opposite of 4x^{4}-25, find the opposite of each term.

$$8\left(-x\right)+2\left(-x\right)x^{2}+13x^{2}+4x\left(-x\right)+\left(-x\right)x^{3}+8x+2x^{3}-4x^{4}+25$$

Multiply 8 and -1 to get -8.

$$-8x+2\left(-1\right)xx^{2}+13x^{2}+4x\left(-1\right)x-xx^{3}+8x+2x^{3}-4x^{4}+25$$

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

$$-8x+2\left(-1\right)x^{3}+13x^{2}+4x\left(-1\right)x-xx^{3}+8x+2x^{3}-4x^{4}+25$$

Multiply 2 and -1 to get -2.

$$-8x-2x^{3}+13x^{2}+4x\left(-1\right)x-xx^{3}+8x+2x^{3}-4x^{4}+25$$

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

$$-8x-2x^{3}+13x^{2}+4x^{2}\left(-1\right)-xx^{3}+8x+2x^{3}-4x^{4}+25$$

Multiply 4 and -1 to get -4.

$$-8x-2x^{3}+13x^{2}-4x^{2}-xx^{3}+8x+2x^{3}-4x^{4}+25$$

Combine 13x^{2} and -4x^{2} to get 9x^{2}.

$$-8x-2x^{3}+9x^{2}-xx^{3}+8x+2x^{3}-4x^{4}+25$$

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

$$-8x-2x^{3}+9x^{2}-x^{4}+8x+2x^{3}-4x^{4}+25$$

Combine -8x and 8x to get 0.

$$-2x^{3}+9x^{2}-x^{4}+2x^{3}-4x^{4}+25$$

Combine -2x^{3} and 2x^{3} to get 0.

$$9x^{2}-x^{4}-4x^{4}+25$$

Combine -x^{4} and -4x^{4} to get -5x^{4}.

$$9x^{2}-5x^{4}+25$$