Finding values to make an equation with a $x^4$ into a perfect square

Question

For what values of p and q is the expression $$x^4+ 6x^3+ 13x^2+ px + q$$ a perfect square? The answer I got didn't seem correct so if someone break it down to me how about going this problem it would be appreciated

Answer 1

Hint: write like this

$$x^4+ 6x^3+ 13x^2+ px + q= (x^2+ ax +b)^2$$

$$=x^4+ 2ax^3+ (2b+a^2)x^2+ 2abx + b^2$$

---

So $a =3$ and $2b+9 = 13$ so $b= 2$. Then $p = 2ab =12$ and $q=b^2 =4$.