Multiply 3x^2 (5x^3)

Step-by-step explanation:

Here's what the multiplication looks like when it's done horizontally:

(4x2 - 4x - 7) (x + 3)

(4x2 - 4x - 7) (x) + (4x2 - 4x - 7) (3)

4x2 (x) - 4x (x) - 7 (x) + 4x2 (3) - 4x (3) - 7 (3)

4x3 - 4x2 - 7x + 12x2 - 12x - 21

4x3 - 4x2 + 12x2 - 7x - 12x - 21

4x3 + 8x2 - 19x - 21

That was painful! Now I'll do it vertically:

4x^2 - 4x - 7 is positioned above x + 3; first row: + 3 times - 7 is - 21, carried down below the + 3; + 3 times - 4x is - 12x, carried down below the x; + 3 times 4x^2 is + 12x^2, carried down to the left of the - 12x; second row: x times - 7 is - 7x, carried down below the - 12x; x times - 4x is - 4x^2, carried down below the + 12x^2; x times 4x^2 is 4x^3, carried down to the left of the - 4x^2; adding down: 4x^3 + (+12x^2) + (-4x^2) + (-12x) + (-7x) + (-21) = 4x^3 + 8x^2 - 19x - 21

That was a lot easier! But, by either method, the answer is the same:

4x3 + 8x2 - 19x - 21

Simplify (x + 2) (x3 + 3x2 + 4x - 17)

I'm just going to do this one vertically; horizontally is too much trouble.

Note that, since order doesn't matter for multiplication, I can still put the "x + 2" polynomial on the bottom for the vertical multiplication, just as I always put the smaller number on the bottom when I was doing regular vertical multiplication with just plain numbers back in grammar school.

x^3 + 3x^2 + 4x - 17 is positioned above x + 2; first row: + 2 times - 17 is - 34, carried down below the + 2; + 2 times + 4x is + 8x, carried down below the x; + 2 times 3x^2 is + 6x^2, carried down to the left of the 8x; + 2 times x^3 is + 2x^3, carried down to the left of the + 6x^2; second row: x times - 17 is - 17x, carried down below the + 8x; x times + 4x is + 4x^2, carried down below the + 6x^2; x times + 3x^2 is + 3x^3, carried down below the + 2x^3; x times x^3 is x^4, carried down to the left of the + 3x^3; adding down: x^4 + (+2x^3) + (3x^3) + (+6x^2) + (+4x^2) + (+8x) + (-17x) + (-34) = x^4 + 5x^3 + 10x^2 - 9x - 34

x4 + 5x3 + 10x2 - 9x - 34

15x^5

Step-by-step explanation:

Multiply the integers and add the powers.

That's

3 x 5 x^2+3

15x^5