Answer
$x=-1,-\frac{1}{4}$.
$f(x) =(4x+1)(x+1)(x^2+2)$
Work Step by Step
Step 1. For $f(x)=4x^4+5x^3+9x^2+10x+2$, list possible rational real zeros $\frac{p}{q}: \pm1,\pm2,\pm\frac{1}{2},\pm\frac{1}{4}$
Step 2. Use synthetic division as shown in the figure to find a zero(s) $x=-1,-\frac{1}{4}$.
Step 3. Use the quotient to find other zeros: $4x^2+8=0$ no real solutions.
Step 4. Factor the polynomial $f(x)=(x+\frac{1}{4})(x+1)(4x^2+8)=(4x+1)(x+1)(x^2+2)$
