Solve {(2x+1)}{(x-3)}{(x-3)}-{(-3x+4)}{(x-3)}{(x+3)}

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

Evaluate

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

Solution Steps

Multiply x-3 and x-3 to get \left(x-3\right)^{2}.

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

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

$$\left(2x+1\right)\left(x^{2}-6x+9\right)-\left(-3x+4\right)\left(x-3\right)\left(x+3\right)$$

Apply the distributive property by multiplying each term of 2x+1 by each term of x^{2}-6x+9.

$$2x^{3}-12x^{2}+18x+x^{2}-6x+9-\left(-3x+4\right)\left(x-3\right)\left(x+3\right)$$

Combine -12x^{2} and x^{2} to get -11x^{2}.

$$2x^{3}-11x^{2}+18x-6x+9-\left(-3x+4\right)\left(x-3\right)\left(x+3\right)$$

Combine 18x and -6x to get 12x.

$$2x^{3}-11x^{2}+12x+9-\left(-3x+4\right)\left(x-3\right)\left(x+3\right)$$

Apply the distributive property by multiplying each term of -3x+4 by each term of x-3.

$$2x^{3}-11x^{2}+12x+9-\left(-3x^{2}+9x+4x-12\right)\left(x+3\right)$$

Combine 9x and 4x to get 13x.

$$2x^{3}-11x^{2}+12x+9-\left(-3x^{2}+13x-12\right)\left(x+3\right)$$

Apply the distributive property by multiplying each term of -3x^{2}+13x-12 by each term of x+3.

$$2x^{3}-11x^{2}+12x+9-\left(-3x^{3}-9x^{2}+13x^{2}+39x-12x-36\right)$$

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

$$2x^{3}-11x^{2}+12x+9-\left(-3x^{3}+4x^{2}+39x-12x-36\right)$$

Combine 39x and -12x to get 27x.

$$2x^{3}-11x^{2}+12x+9-\left(-3x^{3}+4x^{2}+27x-36\right)$$

To find the opposite of -3x^{3}+4x^{2}+27x-36, find the opposite of each term.

$$2x^{3}-11x^{2}+12x+9-\left(-3x^{3}\right)-4x^{2}-27x-\left(-36\right)$$

The opposite of -3x^{3} is 3x^{3}.

$$2x^{3}-11x^{2}+12x+9+3x^{3}-4x^{2}-27x-\left(-36\right)$$

The opposite of -36 is 36.

$$2x^{3}-11x^{2}+12x+9+3x^{3}-4x^{2}-27x+36$$

Combine 2x^{3} and 3x^{3} to get 5x^{3}.

$$5x^{3}-11x^{2}+12x+9-4x^{2}-27x+36$$

Combine -11x^{2} and -4x^{2} to get -15x^{2}.

$$5x^{3}-15x^{2}+12x+9-27x+36$$

Combine 12x and -27x to get -15x.

$$5x^{3}-15x^{2}-15x+9+36$$

Add 9 and 36 to get 45.

$$5x^{3}-15x^{2}-15x+45$$

Expand

$$5x^{3}-15x^{2}-15x+45$$

Solution Steps

Multiply x-3 and x-3 to get \left(x-3\right)^{2}.

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

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

$$\left(2x+1\right)\left(x^{2}-6x+9\right)-\left(-3x+4\right)\left(x-3\right)\left(x+3\right)$$

Apply the distributive property by multiplying each term of 2x+1 by each term of x^{2}-6x+9.

$$2x^{3}-12x^{2}+18x+x^{2}-6x+9-\left(-3x+4\right)\left(x-3\right)\left(x+3\right)$$

Combine -12x^{2} and x^{2} to get -11x^{2}.

$$2x^{3}-11x^{2}+18x-6x+9-\left(-3x+4\right)\left(x-3\right)\left(x+3\right)$$

Combine 18x and -6x to get 12x.

$$2x^{3}-11x^{2}+12x+9-\left(-3x+4\right)\left(x-3\right)\left(x+3\right)$$

Apply the distributive property by multiplying each term of -3x+4 by each term of x-3.

$$2x^{3}-11x^{2}+12x+9-\left(-3x^{2}+9x+4x-12\right)\left(x+3\right)$$

Combine 9x and 4x to get 13x.

$$2x^{3}-11x^{2}+12x+9-\left(-3x^{2}+13x-12\right)\left(x+3\right)$$

Apply the distributive property by multiplying each term of -3x^{2}+13x-12 by each term of x+3.

$$2x^{3}-11x^{2}+12x+9-\left(-3x^{3}-9x^{2}+13x^{2}+39x-12x-36\right)$$

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

$$2x^{3}-11x^{2}+12x+9-\left(-3x^{3}+4x^{2}+39x-12x-36\right)$$

Combine 39x and -12x to get 27x.

$$2x^{3}-11x^{2}+12x+9-\left(-3x^{3}+4x^{2}+27x-36\right)$$

To find the opposite of -3x^{3}+4x^{2}+27x-36, find the opposite of each term.

$$2x^{3}-11x^{2}+12x+9-\left(-3x^{3}\right)-4x^{2}-27x-\left(-36\right)$$

The opposite of -3x^{3} is 3x^{3}.

$$2x^{3}-11x^{2}+12x+9+3x^{3}-4x^{2}-27x-\left(-36\right)$$

The opposite of -36 is 36.

$$2x^{3}-11x^{2}+12x+9+3x^{3}-4x^{2}-27x+36$$

Combine 2x^{3} and 3x^{3} to get 5x^{3}.

$$5x^{3}-11x^{2}+12x+9-4x^{2}-27x+36$$

Combine -11x^{2} and -4x^{2} to get -15x^{2}.

$$5x^{3}-15x^{2}+12x+9-27x+36$$

Combine 12x and -27x to get -15x.

$$5x^{3}-15x^{2}-15x+9+36$$

Add 9 and 36 to get 45.

$$5x^{3}-15x^{2}-15x+45$$