Question 1114140
<br>
Since the leading term is 16x^4 and the constant term is 49, the perfect square must be of the form {{{(4x^2+nx+7)^2}}} or {{{4x^2+nx-7)^2}}}.<br>
Each form gives an answer to the question.<br>
(1) For the first form...<br>
{{{(4x^2+nx+7)^2 = 16x^4+8nx^3+(n^2+56)x^2+14nx+49}}}<br>
Then
8n = -24  -->  n = -3
n^2+56 = 65 = a-1  -->  a = 66
14n = -42 = b+1  -->  b = -43<br>
(2) For the second form...<br>
{{{(4x^2+nx-7)^2 = 16x^4+8nx^3+(n^2-56)x^2-14nx+49}}}<br>
Then
8n = -24  -->  n = -3
n^2-56 = -47 = a-1  -->  a = -46
-14n = 42 = b+1  -->  b = 41<br>
Answer: Two solutions
(1) a=66, b=-43
(2) a=-46, b=41