Solve aleft(x-3right)^2+left(2-aright)left(x-3right)-aleft(x+3right)^2+left(2-aright)left(x+3right)

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

Evaluate

$$4x-14ax$$

Solution Steps

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

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

Use the distributive property to multiply a by x^{2}-6x+9.

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

Use the distributive property to multiply 2-a by x-3.

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

Combine -6ax and -ax to get -7ax.

$$ax^{2}-7ax+9a+2x-6+3a-a\left(x+3\right)^{2}+\left(2-a\right)\left(x+3\right)$$

Combine 9a and 3a to get 12a.

$$ax^{2}-7ax+12a+2x-6-a\left(x+3\right)^{2}+\left(2-a\right)\left(x+3\right)$$

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

$$ax^{2}-7ax+12a+2x-6-a\left(x^{2}+6x+9\right)+\left(2-a\right)\left(x+3\right)$$

Use the distributive property to multiply a by x^{2}+6x+9.

$$ax^{2}-7ax+12a+2x-6-\left(ax^{2}+6ax+9a\right)+\left(2-a\right)\left(x+3\right)$$

To find the opposite of ax^{2}+6ax+9a, find the opposite of each term.

$$ax^{2}-7ax+12a+2x-6-ax^{2}-6ax-9a+\left(2-a\right)\left(x+3\right)$$

Combine ax^{2} and -ax^{2} to get 0.

$$-7ax+12a+2x-6-6ax-9a+\left(2-a\right)\left(x+3\right)$$

Combine -7ax and -6ax to get -13ax.

$$-13ax+12a+2x-6-9a+\left(2-a\right)\left(x+3\right)$$

Combine 12a and -9a to get 3a.

$$-13ax+3a+2x-6+\left(2-a\right)\left(x+3\right)$$

Use the distributive property to multiply 2-a by x+3.

$$-13ax+3a+2x-6+2x+6-ax-3a$$

Combine 2x and 2x to get 4x.

$$-13ax+3a+4x-6+6-ax-3a$$

Add -6 and 6 to get 0.

$$-13ax+3a+4x-ax-3a$$

Combine 3a and -3a to get 0.

$$-13ax+4x-ax$$

Combine -13ax and -ax to get -14ax.

$$-14ax+4x$$

Expand

$$4x-14ax$$

Solution Steps

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

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

Use the distributive property to multiply a by x^{2}-6x+9.

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

Use the distributive property to multiply 2-a by x-3.

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

Combine -6ax and -ax to get -7ax.

$$ax^{2}-7ax+9a+2x-6+3a-a\left(x+3\right)^{2}+\left(2-a\right)\left(x+3\right)$$

Combine 9a and 3a to get 12a.

$$ax^{2}-7ax+12a+2x-6-a\left(x+3\right)^{2}+\left(2-a\right)\left(x+3\right)$$

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

$$ax^{2}-7ax+12a+2x-6-a\left(x^{2}+6x+9\right)+\left(2-a\right)\left(x+3\right)$$

Use the distributive property to multiply a by x^{2}+6x+9.

$$ax^{2}-7ax+12a+2x-6-\left(ax^{2}+6ax+9a\right)+\left(2-a\right)\left(x+3\right)$$

To find the opposite of ax^{2}+6ax+9a, find the opposite of each term.

$$ax^{2}-7ax+12a+2x-6-ax^{2}-6ax-9a+\left(2-a\right)\left(x+3\right)$$

Combine ax^{2} and -ax^{2} to get 0.

$$-7ax+12a+2x-6-6ax-9a+\left(2-a\right)\left(x+3\right)$$

Combine -7ax and -6ax to get -13ax.

$$-13ax+12a+2x-6-9a+\left(2-a\right)\left(x+3\right)$$

Combine 12a and -9a to get 3a.

$$-13ax+3a+2x-6+\left(2-a\right)\left(x+3\right)$$

Use the distributive property to multiply 2-a by x+3.

$$-13ax+3a+2x-6+2x+6-ax-3a$$

Combine 2x and 2x to get 4x.

$$-13ax+3a+4x-6+6-ax-3a$$

Add -6 and 6 to get 0.

$$-13ax+3a+4x-ax-3a$$

Combine 3a and -3a to get 0.

$$-13ax+4x-ax$$

Combine -13ax and -ax to get -14ax.

$$-14ax+4x$$