Step 1:

Use rational root test to find out that the $x=1$ is a root of polynomial $x^4-5x^2+4$.

The Rational Root Theorem tells us that if the polynomial has a rational zero then it must be a fraction $\frac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

The constant term is $4$, with a single factor of 1, 2 and 4.

The leading coefficient is $1$, with a single factor of 1.

The POSSIBLE zeroes are:

Substitute the possible roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.

We can see that $p\left(1\right)=0$ so $x=1$ is a root of a polynomial $p(x)$.

To find remaining zeros we use Factor Theorem. This theorem states that if $\frac{p}{q}$ is root of the polynomial then the polynomial can be divided by $qx-p$. In this example we divide polynomial $p$ by $x-1$

Step 2:

The next rational root is $x=1$

Step 3:

The next rational root is $x=2$

Step 4:

The next rational root is $x=-1$

Step 5:

To find the last zero, solve equation $x+2=0$