Reducible

> f:=(p,q)->(p+5*q+5)^2+(2*p+q-19/72):expand(f(p,q));ToCanonical(f);

p^2+10*p*q+12*p+25*q^2+51*q+1781/72

[p^2+q, ` case (6) `]

> f:=(p,q)->(p+5*q+5)^2+(2*p+q-19/72)^2+168:expand(f(p,q));ToCanonical(f);

5*p^2+14*p*q+161/18*p+26*q^2+1781/36*q+1000873/5184

[p^2+q^2+1, ` case (7) `]

> f:=(p,q)->(p+5*q+5)*(2*p+q-19/72)*168:expand(f(p,q));ToCanonical(f);

336*p^2+1848*p*q+4907/3*p+840*q^2+1855/3*q-665/3

[p*q, ` case (8) `]

> f:=(p,q)->(p+5*q+5)*(2*p+q)*(3*p+6*q+5):expand(f(p,q));ToCanonical(f);

6*p^3+45*p^2*q+40*p^2+81*p*q^2+130*p*q+30*q^3+55*q^2+50*p+25*q

`equivalent either to a binary form (case (9) or (10) or to p^3 (case (11))`

>

> f:=(p,q)->(p+5*q+5)^2+(2*p+q-19)^2+1:expand(f(p,q));ToCanonical(f);

5*p^2+14*p*q-66*p+26*q^2+12*q+387

[p^2+q^2+1, ` case (7) `]

> f:=(p,q)->(p+5*q+5)*(2*p+q-19):expand(f(p,q));ToCanonical(f);

2*p^2+11*p*q-9*p+5*q^2-90*q-95

[p*q, ` case (8) `]

>