$\displaystyle Multiply \; (3 x-2) (5 x^2+x+1)\; using \; a \; grid$

$\displaystyle Answer: \; 15 x^3-7 x^2+x-2$

To multiply two binomials, both terms in the first binomial need to be multiplied with both terms in the second binomial. A good way to make sure that all of the pairs are multiplied is to set up the box/grid:

In each empty box, multiply the intersecting terms:

Now, combine like terms. 3x and 8x are both terms with x, so we can add them to be 11x.

Final answer [in descending order of powers of x:

$\displaystyle -x^2 +11x -24$

To multiply two binomials, both terms in the first binomial need to be multiplied with both terms in the second binomial. A good way to make sure that all of the pairs are multiplied is to set up the box/grid:

In each empty box, multiply the intersecting terms: 

Now, combine like terms. -6x and 4x are both terms with x, so we can add them to be -2x.

Final answer: $\displaystyle 2x^2 -2x -12$

Top left corner:

 $\displaystyle x*x=x^2$

Top right corner:

$\displaystyle x*-6=-6x$

Bottom left corner:

$\displaystyle +3*x=3x$

Bottom right corner:

$\displaystyle +3*-6=-18$

Add along the diagonal:

$\displaystyle -6x+3x=-3x$

Combining our terms:

$\displaystyle x^2-3x-18$

Top left corner:

 $\displaystyle 3x*x=3x^2$

Top right corner:

$\displaystyle 3x*-9=-27x$

Bottom left corner:

$\displaystyle 4*x=4x$

Bottom right corner:

$\displaystyle 4*-9=-36$

Add along the pink diagonal:

$\displaystyle 4x-27x=-23x$

Combining our terms:

$\displaystyle 3x^2-23x-36$

Top left corner:

 $\displaystyle 5x^2*3x=15x^3$

Top right corner:

$\displaystyle 5x^2*-2=-10x^2$

Middle left:

$\displaystyle x*3x=3x^2$

Middle right:

$\displaystyle x*-2=-2x$

Bottom left corner:

$\displaystyle 1*3x=3x$

Bottom right corner:

$\displaystyle 1*-2=-2$

Add along the first (pink) diagonal:

$\displaystyle 3x^2-10x^2=-7x^2$

Add along the second (blue) diagonal:

$\displaystyle 3x-2x=x$

Combining our terms:

$\displaystyle 15x^3-7x^2+x-2$

Top left corner:

 $\displaystyle x*-x=-x^2$

Top right corner:

$\displaystyle x*1=x$

Bottom left corner:

$\displaystyle -6*-x=6x$

Bottom right corner:

$\displaystyle -6*1=-6$

Add along the (pink) diagonal:

$\displaystyle 6x+x=7x$

Combining our terms:

$\displaystyle -x^2+7x-6$

Top left corner:

 $\displaystyle -x*3x=-3x^2$

Top right corner:

$\displaystyle -x*-2=2x$

Bottom left corner:

$\displaystyle 2*3x=6x$

Bottom right corner:

$\displaystyle 2*-2=-4$

Add along the (pink) diagonal:

$\displaystyle 6x+2x=8x$

Combining our terms:

$\displaystyle -3x^2+8x-4$

Let's look at the grid for $\displaystyle (2x+4)(x-3)$ first.

Now let's look at the grid for $\displaystyle (x-3)(2x+4)$.

Top left corner:

 $\displaystyle 2x*x=2x^2$ or $\displaystyle x*2x=2x^2$

Both grids give us $\displaystyle 2x^2$.

Top right corner:

$\displaystyle 2x*-3=-6x$ or $\displaystyle x*4=4x$

Bottom left corner:

$\displaystyle 4*x=4x$ or $\displaystyle -3*2x=-6x$

Bottom right corner:

$\displaystyle 4*-3=-12$ or $\displaystyle -3*4=-12$

Both grids give us $\displaystyle -12$ here.

Add along the (pink) diagonals in each grid:

$\displaystyle 4x-6x=-2x$ or $\displaystyle -6x+4x=-2x$

Both grids give us $\displaystyle -2x$.

Combining our terms in either grid, we get:

$\displaystyle 2x^2-2x-12$

So the answers are exactly the same!

Top left corner:

 $\displaystyle -10x*x=-10x^2$

Top right corner:

$\displaystyle -10x*3=-30x$

Bottom left corner:

$\displaystyle -1*x=-x$ 

Bottom right corner:

$\displaystyle -1*3=-3$ 

Add along the (pink) diagonal:

$\displaystyle -x-30x=-31x$

Combining our terms, we get:

$\displaystyle -10x^2-31x-3$