Answer:
2, -2, and 0.
Step-by-step explanation:
First, we need to factorize the left side of the equation as:
(x² + 4)² + 32 = 12x² + 48
(x² + 4)² + 32 = 12( x² + 4)
Now, we need to substitute a = x² + 4, so:
a² + 32 = 12a
So, solving for a, we get:
a² + 32 - 12a = 12a - 12a
a² + 32 - 12a = 0
a² - 12a + 32 = 0
(a - 8)(a - 4) = 0
Then:
a - 8 = 0
a - 8 + 8 = 0 + 8
a = 8
or
a - 4 = 0
a - 4 + 4 = 0 + 4
a = 4
Then, a is equal to x² + 4, so replacing a by x² + 4 on each solution for a, we get:
For a = 8:
x² + 4 = 8
x² + 4 - 4 = 8 - 4
x² = 4
x = ±√4
x = ± 2
For a = 4:
x² + 4 = 4
x² + 4 - 4 = 4 - 4
x² = 0
x = 0
Therefore, the solutions for x are 2, -2, and 0.