Question

How do you multiply `\frac { x - 4} { 4x ^ { 2} + 10x } \cdot \frac { 4x ^ { 3} + 10x ^ { 2} } { 2x ^ { 2} }`?

Answer 1

`(x-4)/(2x)`

Explanation:

When multiplying fractions we simply multiply across.
Let's start with the numerators.

`(x-4)(4x^3+10x^2)`

Remember we multiply every term with each other.
So,

`(x*4x^3)+(x*10x^2)+(-4*4x^3)+(-4*10x^2)`

`4x^4+10x^3-16x^3-40x^2`

[Simplify common variables]

`4x^4color(red)(+10x^3-16x^3)-40x^2`

`4x^4-6x^3-40x^2`

`color(blue)"Now we do the same for the denominators"`

`(4x^2+10x)(2x^2)`

`(4x^2*2x^2)+(10x*2x^2)`

`8x^4+20x^3`

So now we have...
our numerator
`4x^4-6x^3-40x^2`

and our denominator
`8x^4+20x^3`

`(4x^4-6x^3-40x^2)/(8x^4+20x^3)`

Now look for common factors to simplify our fraction.
I found `2x^2` in the numerator and `4x^3` in the denominator

`(2x^2(2x^2-3x-20))/(4x^3(2x+5))`

Divide our common factor

`((2x^2-3x-20))/(2x(2x+5))`

`(2x^2-3x-20)/(2x(2x+5)`

Factor the numerator using whatever method your prefer.
Look here if you are confused on how I factored the numerator.
Your result should be

`((x-4)(2x+5))/(2x(2x+5)`

`((x-4)cancel((2x+5)))/(2xcancel((2x+5))`

`(x-4)/(2x)`