Pell Equations: $a^2+4=5b^2$

Question

This is a challenge problem in the Pell Equations chapter of my number theory book, but I'm not seeing the connection to Pell Equations. The Pell Equation with the coefficient $5$ is $5b^2+1=a^2$, but it doesn't look like the one I have.

Thanks if you can help me.

Answer 1

if a,b are natural numbers :$$a-b<a+b\\(a-b)(a+b)=4(b-1)(b+1)=\\1*(4(b-1)(b+1)\\=2*2(b-1)(b+1)\\=(b-1)*(4(b+1)\\=(b+1)*(4(b-1))\\=2(b-1)*(2(b+1))$$now try them to get answer