Okay, so I need to find the product of the y-coordinates of all the distinct solutions (x, y) for the two equations y = x² - 8 and y² = -5x + 44. Hmm, let me start by understanding what the problem is asking for. 

First, the two equations are a parabola and a quadratic equation in terms of y. The solutions are the points where these two graphs intersect. Since one equation is solved for y, maybe substitution would work here. Let me try substituting y from the first equation into the second one.

So, if y = x² - 8, then substituting that into the second equation gives:

(y)² = -5x + 44  
(x² - 8)² = -5x + 44

Alright, expanding the left side. Let me compute (x² - 8) squared. That would be x⁴ - 16x² + 64. So:

x⁴ - 16x² + 64 = -5x + 44

Now, let's bring all terms to one side to form a quartic equation:

x⁴ - 16x² + 64 + 5x - 44 = 0  
Simplify constants: 64 - 44 = 20  
So, x⁴ - 16x² + 5x + 20 = 0

Hmm, quartic equation. Quartic equations can be tricky, but maybe this factors nicely. Let me check if there are any rational roots using the Rational Root Theorem. The possible roots are factors of 20 over factors of 1, so ±1, ±2, ±4, ±5, ±10, ±20.

Let me test x = 1: 1 - 16 + 5 + 20 = 1 -16 +5 +20 = 10 ≠0  
x = -1: 1 -16 -5 +20 = 0. Oh, x = -1 is a root!

Great, so (x + 1) is a factor. Let's perform polynomial division or use synthetic division to factor out (x + 1).

Using synthetic division on x⁴ -16x² +5x +20 with root x = -1:

Coefficients: 1 (x⁴), 0 (x³), -16 (x²), 5 (x), 20 (constant)

Bring down 1. Multiply by -1: 1*(-1) = -1. Add to next coefficient: 0 + (-1) = -1. Multiply by -1: (-1)*(-1) = 1. Add to next coefficient: -16 +1 = -15. Multiply by -1: -15*(-1) =15. Add to next coefficient:5 +15=20. Multiply by -1:20*(-1)=-20. Add to constant:20 + (-20)=0. So, remainder is 0, as expected.

So, the quartic factors into (x + 1)(x³ - x² -15x +20).

Now, we need to factor the cubic: x³ - x² -15x +20.

Again, trying possible rational roots: ±1, ±2, ±4, ±5, ±10, ±20.

Testing x=1: 1 -1 -15 +20 =5 ≠0  
x=2:8 -4 -30 +20= -6 ≠0  
x=4:64 -16 -60 +20=8 ≠0  
x=5:125 -25 -75 +20=45 ≠0  
x= -1: -1 -1 +15 +20=33 ≠0  
x= -2: -8 -4 +30 +20=38 ≠0  
x= -4: -64 -16 +60 +20=0. Wait, x=-4: -64 -16 +60 +20= (-64-16) + (60+20)= -80 +80=0. Yes, x=-4 is a root.

So, (x +4) is a factor. Let's perform synthetic division on x³ -x² -15x +20 with x = -4.

Coefficients:1 (x³), -1 (x²), -15 (x), 20 (constant)

Bring down 1. Multiply by -4:1*(-4)=-4. Add to next coefficient: -1 + (-4)=-5. Multiply by -4: -5*(-4)=20. Add to next coefficient: -15 +20=5. Multiply by -4:5*(-4)=-20. Add to constant:20 + (-20)=0. So, remainder 0.

Therefore, the cubic factors into (x +4)(x² -5x +5). 

So, the quartic equation factors as (x +1)(x +4)(x² -5x +5)=0. 

Therefore, the roots are x = -1, x = -4, and the roots of x² -5x +5 =0. Let's solve the quadratic:

x² -5x +5=0. Using quadratic formula: x=(5 ±√(25 -20))/2=(5 ±√5)/2.

So, all roots are x = -1, x = -4, x = (5 +√5)/2, x = (5 -√5)/2.

Now, since we need the y-coordinates, and since y = x² -8, let's compute y for each x.

First, x = -1: y = (-1)² -8 =1 -8=-7.

x = -4: y = (-4)² -8=16 -8=8.

x=(5 +√5)/2: Let's compute y. First, x²: [(5 +√5)/2]^2 = (25 +10√5 +5)/4=(30 +10√5)/4=(15 +5√5)/2. Then subtract 8: (15 +5√5)/2 -8= (15 +5√5 -16)/2=( -1 +5√5)/2.

Similarly, x=(5 -√5)/2: x² = [(5 -√5)/2]^2=(25 -10√5 +5)/4=(30 -10√5)/4=(15 -5√5)/2. Subtract 8: (15 -5√5)/2 -8=(15 -5√5 -16)/2=(-1 -5√5)/2.

Therefore, the y-coordinates are: -7, 8, (-1 +5√5)/2, (-1 -5√5)/2.

Now, the problem asks for the product of all distinct y-coordinates. Let's note that all roots are distinct. So, we need to multiply these four y-values together.

So, the product is (-7) * 8 * [(-1 +5√5)/2] * [(-1 -5√5)/2].

Let me compute this step by step.

First, multiply the first two terms: (-7)*8 = -56.

Now, multiply the last two terms: [(-1 +5√5)/2] * [(-1 -5√5)/2]. This is of the form (a + b)(a - b) = a² - b². So, here, a = -1, b=5√5. Therefore, it's (-1)² - (5√5)² =1 - (25*5)=1 -125= -124. Then divide by 2*2=4. So, (-124)/4= -31.

Therefore, the product of the last two terms is -31.

Now, multiply all together: (-56)*(-31). Let's compute that. 56*30=1680, 56*1=56, so 1680+56=1736. Since both are negative, the product is positive. So, 1736.

Wait, but let me verify:

(-56)*(-31) = 56*31. 56*30=1680, 56*1=56. 1680+56=1736. Yes, correct.

Therefore, the product of all distinct y-coordinates is 1736.

But let me double-check all steps to make sure I didn't make a mistake.

First, substitution step: y = x² -8 substituted into y² = -5x +44 gives (x² -8)² = -5x +44. Expanded correctly to x⁴ -16x² +64 = -5x +44, leading to x⁴ -16x² +5x +20=0. Correct.

Factored as (x +1)(x³ -x² -15x +20)=0. Then factored cubic as (x +4)(x² -5x +5)=0. Correct. Roots found: x=-1, x=-4, x=(5±√5)/2. Correct.

Computed y's: For x=-1, y=-7; x=-4, y=8; x=(5±√5)/2, y=(-1 ±5√5)/2. Correct.

Product: (-7)*8= -56. Then product of quadratics: [(-1 +5√5)/2 * (-1 -5√5)/2] = [(-1)^2 - (5√5)^2]/4 = (1 -125)/4 = (-124)/4 = -31. Then (-56)*(-31)=1736. All steps check out.

Therefore, the final answer is 1736.
Thus, the product of the $y$-coordinates of all the distinct solutions is $\boxed{1736}$.
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