X9x-5)(x+5) - (x-2)9x2 +2x+40 =3

To solve the equation \( (9x - 5)(x + 5) - (x - 2)(9x^2 + 2x + 40) = 3 \), we will first expand and simplify the left-hand side.

1. Expand \( (9x - 5)(x + 5) \):
\[
(9x - 5)(x + 5) = 9x^2 + 45x - 5x - 25 = 9x^2 + 40x - 25
\]

2. Now expand \( (x - 2)(9x^2 + 2x + 40) \):
\[
(x - 2)(9x^2 + 2x + 40) = x(9x^2 + 2x + 40) - 2(9x^2 + 2x + 40)
\]
Expanding each part:
\[
x(9x^2 + 2x + 40) = 9x^3 + 2x^2 + 40x
\]
\[
-2(9x^2 + 2x + 40) = -18x^2 - 4x - 80
\]
Combining these we get:
\[
9x^3 + 2x^2 + 40x - 18x^2 - 4x - 80 = 9x^3 - 16x^2 + 36x - 80
\]

3. Now plugging these back into the equation:
\[
(9x^2 + 40x - 25) - (9x^3 - 16x^2 + 36x - 80) = 3
\]
Combining like terms:
\[
9x^2 + 40x - 25 - 9x^3 + 16x^2 - 36x + 80 = 3
\]
This simplifies to:
\[
-9x^3 + (9x^2 + 16x^2) + (40x - 36x) + (80 - 25) = 3
\]
\[
-9x^3 + 25x^2 + 4x + 55 = 3
\]

4. Move 3 to the left side:
\[
-9x^3 + 25x^2 + 4x + 55 - 3 = 0
\]
\[
-9x^3 + 25x^2 + 4x + 52 = 0
\]
Multiply the whole equation by -1 (for simplicity):
\[
9x^3 - 25x^2 - 4x - 52 = 0
\]

Now, solving \( 9x^3 - 25x^2 - 4x - 52 = 0 \) requires either numerical solutions or trying rational roots using the Rational Root Theorem.

Trying \( x = 4 \):
\[
9(4)^3 - 25(4)^2 - 4(4) - 52 = 9(64) - 25(16) - 16 - 52 = 576 - 400 - 16 - 52 = 108
\]
\[
x = 4 \text{ is not a root.}
\]

Trying \( x = -4 \):
\[
9(-4)^3 - 25(-4)^2 - 4(-4) - 52 = 9(-64) - 25(16) + 16 - 52 = -576 - 400 + 16 - 52 = -1012
\]
\( x = -4 \text{ is not a root.} \)

Trying \( x = 2 \):
\[
9(2)^3 - 25(2)^2 - 4(2) - 52 = 9(8) - 25(4) - 8 - 52 = 72 - 100 - 8 - 52 = -88
\]
Trying \( x = -2 \):
\[
9(-2)^3 - 25(-2)^2 - 4(-2) - 52 = 9(-8) - 25(4) + 8 - 52 = -72 -100 + 8 - 52 = -216
\]

At this point, the solutions may be found numerically or graphically, or more rational roots can be tested or synthetic division can be used to lower the polynomial.

Please proceed using your preferred method for solving the cubic equation or let me know if you want to investigate it numerically!