Solution via Context Panel

$y\prime \=4x{y}^2\+8y^2\+x\+2$

$\frac{\ⅆ}{\ⅆx}y\left(x\right)\=4x{y\left(x\right)}^2\+8{y\left(x\right)}^2\+x\+2$

$\stackrel{\text{add initial condition}}{\to }$

$\frac{\ⅆ}{\ⅆx}y\left(x\right)\=4x{y\left(x\right)}^2\+8{y\left(x\right)}^2\+x\+2\,y\left(1\right)\=0$

$\stackrel{\text{solve DE}}{\to }$

$y\left(x\right)\=\frac{1}{2}\tan \left(x^2\+4x-5\right)$

$\frac{1}{2}\tan \left(x^2\+4x-5\right)$$\to$

Factor the right-hand side of the differential equation (a prelude to separating variables)

$4 x y^2\+8y^2\+x\+2$$\stackrel{\text{factor}}{\=}$$\left(4y^2\+1\right)\left(x\+2\right)$