Question

How do you factor `2x^4+5x^2+3`?

Answer 1

`2x^4+5x^2+3=(x^2+1)(2x^2+3)`

Explanation:

Don't get scared by the fact that there is a fourth power of `x`. The only thing to notice here is that only powers of `x^2` are present in the expression. Let's use that to our advantage.

Assume `x^2=a`. Substituting this into our expression, we get
`2x^4+5x^2+3=2a^2+5a+3`.

The problem is now basically that of middle term factorization. We notice that `2a^2+5a+3=2a^2+2a+3a+3=2a(a+1)+3(a+1)=(a+1)(2a+3)`.

But we're not done yet! We must put back the substituted value of `a`, i.e. `x^2`. Doing so, we get:

`2x^4+5x^2+3=(x^2+1)(2x^2+3)`