Find Integral values of x for which x² + 19x + 46 is a perfect square?

 x²+19x+46

=x²+18x+81+x-35

=(x+9)²+(x-35)

for the above equation to become perfect square x-35 must be equal to zero

therefore x=35

similarly, x²+19x+46=x²+20x+100-(x+54)

=(x+10)²-(x+54)

here x+54=0

=>  x=-54

x²+19x+46=x²+22x+121-3(x+25)

=(x+11)²-3(x+25)

here  x+25=0

=>  x=-25

x²+19x+46=x²+16x+64+3x-18

=(x+8)²+3(x-6)

here  x-6=0

=>  x=6