Question

How do you factor by grouping: `3x^2 + 3 + x^3 + x`?

Answer 1

The factored form of the expression is `(x^2+1)(x+3)`.

Explanation:

First, rearrange the terms. Then, factor `x^2` out from the first two terms and `1` out from the last two terms. Lastly, combine the two factors to get an answer:

`color(white)=3x^2+3+x^3+x`

`=x^3+3x^2+x+3`

`=color(red)(x^2)*x+color(red)(x^2)*3+x+3`

`=color(red)(x^2)(x+3)+x+3`

`=color(red)(x^2)(x+3)+color(blue)1*x+color(blue)1*3`

`=color(red)(x^2)(x+3)+color(blue)1(x+3)`

`=(color(red)(x^2)+color(blue)1)(x+3)`

Answer 2

This does it: `(x+3)(x^2 + 1)`

Explanation:

I am not sure about the "by grouping" method, but I stumbled on a way to factor that expression.

I noticed that it could be written `3*(x^2 +1) + x*(x^2 +1)`.

From there I put the 3 and the x inside a second set of parentheses and saw that

`3x^2 + 3 + x^3+ x = (3+x)(x^2 + 1)`

So `(3+x)and(x^2 + 1)` are the factors.

A preferable way to write it would probably be `(x+3)(x^2 + 1)`.

I hope this helps,
Steve