In Problems 39–56, find the real zeros of f. Use the real zeros to factor f.

$f\left(x\right)=4x^5+12x^4-x-3$

The given function is $f\left(x\right)=4x^5+12x^4-x-3$

The given function is of degree five, so it has at most five real zeros.

As all the coefficients are integers so we use the rational zeros theorem.

The factors of the constant term$-3$ are

$p:\pm 1,\pm 3$

The factors of the leading coefficient $4$are

$q:\pm 1,\pm 2,\pm 4$

So, the possible rational zeros are

$\frac{p}{q}:\pm 1,\pm 3,\pm \frac{1}{2},\pm \frac{3}{2},\pm \frac{1}{4},\pm \frac{3}{4}$

The graph is

From the graph, we conclude that it has three roots.

Since $-3$appears to be zero and a potential rational zero also.

By evaluating we get,$f(-3)=0$

Thus, $(x+3)$ is a factor of f.

Use synthetic division to factor f

So, factor fis$(x+3)(4x^4-1)$

Put depressed equation to zero and factor by grouping

$$\begin{gathered} 4x^4-1=0 \\ 4x^4=1 \\ {x}^4=\frac{1}{4} \\ x=\pm \frac{1}{\sqrt{2}} \end{gathered}$$

Therefore, the other zeros are$\frac{1}{\sqrt{2}}and-\frac{1}{\sqrt{2}}.$

The factored form of f is$(2x^2+1)(\sqrt{2}x+1)\left(\sqrt{2}x-1\right)(x+3).$