# How do you find the product of (2x^2-2x-3)(2x^2-4x+3)?

Feb 6, 2017

$4 {x}^{4} - 12 {x}^{3} + 8 {x}^{2} + 6 x - 9$

#### Explanation:

When you find the product of anything, all you're doing is multiplying together whatever you're asked to find the product of. In this instance, you're asked to find the product of two polynomials. The easiest way to do this is to use the FOIL method.

When doing this, watch your signs carefully. We will begin by multiplying the first two terms of both trinomials together. Don't forget that when you multiply similar variables with exponents, you add the exponents together. (${x}^{m} \cdot {x}^{n}$ is ${x}^{m + n}$)

$2 {x}^{2} \cdot 2 {x}^{2}$ --> $4 {x}^{4}$

Now continue using the FOIL method. Multiply the first term of the first trinomial by the second term of the second trinomial.

$2 {x}^{2} \cdot - 4 x$ --> $- 8 {x}^{3}$

Continue doing this for the rest of the integers in the trinomials. A method of thinking when FOILing is "everything gets multiplied by everything." Multiply the first term of the first trinomial by the third term of the second trinomial.

$2 {x}^{2} \cdot 3$ --> $6 {x}^{2}$

Now follow this same process all the way through. Multiply the second term of the first trinomial by the first term of the second trinomial, then the second term, and then the third. Repeat this process for the third term of the first trinomial.

$- 2 x \cdot 2 {x}^{2}$ --> $- 4 {x}^{3}$
$- 2 x \cdot - 4 x$ --> $8 {x}^{2}$
$- 2 x \cdot 3$ --> $- 6 x$
$- 3 \cdot 2 {x}^{2}$ --> $- 6 {x}^{2}$
$- 3 \cdot - 4 x$ --> $12 x$
$- 3 \cdot 3$ --> $- 9$

Now, put everything that you've multiplied together.

$4 {x}^{4} - 8 {x}^{3} + 6 {x}^{2} - 4 {x}^{3} + 8 {x}^{2} - 6 x - 6 {x}^{2} + 12 x - 9$

It helps if you put like terms beside each other to make simplifying quicker and less accident-prone.

$4 {x}^{4} - 8 {x}^{3} - 4 {x}^{3} + 6 {x}^{2} + 8 {x}^{2} - 6 {x}^{2} + 12 x - 6 x - 9$

Now just simply and condense all like terms into one expression. This will give you your answer for the product of two trinomials.

$\cancel{4 {x}^{4}} \cancel{- 8 {x}^{3} - 4 {x}^{3}} \cancel{6 {x}^{2} + 8 {x}^{2} - 6 {x}^{2}} \cancel{12 x - 6 x} \cancel{9}$

All like terms combined results in:

$4 {x}^{4} - 12 {x}^{3} + 8 {x}^{2} + 6 x - 9$