3+ [ Completing the square. Sums of squares $]$

What should the values of $a$ and $b$ be so that the polynomial $x^{4}+x^{3}+2 x^{2}+a x+b$ is a perfect square?
The given polynomial of the fourth degree can only be the square of a reduced quadratic trinomial. Therefore, $x^{4}+x^{3}+2 x^{2}+a x+b=\left(x^{2}+p x+q\right)^{2}$. By equating the coefficients of like powers of the argument in both parts of the identity, we get

$2 p=1, p^{2}+2 q=2,2 p q=a, q^{2}=b$. Solving this system of equations, we find $p=1 / 2, q=a=7 / 8, b=49 / 64$.

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