$$ \left(x-7\right)^3+\left(3x+5\right)^3\left(2-4\right)^3=0 $$

Let's break down and solve the given equation step by step.

The equation is:

\[
\left(x - 7\right)^3 + \left(3x + 5\right)^3\left(2 - 4\right)^3 = 0
\]

### Step 1: Simplify the constants
First, simplify $(2 - 4)^3$:
\[
(2 - 4)^3 = (-2)^3 = -8
\]

Now the equation becomes:
\[
(x - 7)^3 + (3x + 5)^3 \cdot (-8) = 0
\]
\[
(x - 7)^3 - 8(3x + 5)^3 = 0
\]

### Step 2: Let $a = (x - 7)$ and $b = (3x + 5)$
This will make the equation look simpler:
\[
a^3 - 8b^3 = 0
\]

### Step 3: Apply the difference of cubes identity
The identity for $a^3 - 8b^3$ is:
\[
a^3 - 8b^3 = (a - 2b)(a^2 + 2ab + 4b^2)
\]

Substitute $a = (x - 7)$ and $b = (3x + 5)$:
\[
\left( (x - 7) - 2(3x + 5) \right) \left( (x - 7)^2 + 2(x - 7)(3x + 5) + 4(3x + 5)^2 \right) = 0
\]

### Step 4: Simplify the first factor
Simplify the expression inside the first factor:
\[
(x - 7) - 2(3x + 5) = x - 7 - 6x - 10 = -5x - 17
\]

So the equation becomes:
\[
(-5x - 17) \left( (x - 7)^2 + 2(x - 7)(3x + 5) + 4(3x + 5)^2 \right) = 0
\]

### Step 5: Solve the first factor
Set $-5x - 17 = 0$:
\[
-5x = 17
\]
\[
x = -\frac{17}{5}
\]

This is one solution. Now, let's check the second factor to see if there are any other solutions.

### Step 6: Expand and solve the second factor
We now expand and simplify the second factor:
\[
(x - 7)^2 + 2(x - 7)(3x + 5) + 4(3x + 5)^2
\]

Let's first expand each term separately:

- $(x - 7)^2 = x^2 - 14x + 49$
- $2(x - 7)(3x + 5) = 2(3x^2 - 16x - 35) = 6x^2 - 32x - 70$
- $4(3x + 5)^2 = 4(9x^2 + 30x + 25) = 36x^2 + 120x + 100$

Now add these up:
\[
x^2 - 14x + 49 + 6x^2 - 32x - 70 + 36x^2 + 120x + 100
\]
\[
= (1x^2 + 6x^2 + 36x^2) + (-14x - 32x + 120x) + (49 - 70 + 100)
\]
\[
= 43x^2 + 74x + 79
\]

Thus, the second factor becomes:
\[
43x^2 + 74x + 79 = 0
\]

### Step 7: Solve the quadratic equation
We now solve the quadratic equation $43x^2 + 74x + 79 = 0$ using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where $a = 43$, $b = 74$, and $c = 79$.

First, calculate the discriminant:
\[
b^2 - 4ac = 74^2 - 4(43)(79)
\]
\[
= 5476 - 13576 = -8100
\]

Since the discriminant is negative, there are no real solutions for this quadratic equation.

### Final Answer:
The only real solution is:
\[
x = -\frac{17}{5}
\]

Would you like any details or have any further questions?

Here are some related questions for further practice:
1. How do you apply the difference of cubes formula in other contexts?
2. Can you explain why a negative discriminant means no real solutions?
3. How would you solve $(a^3 - 27b^3) = 0$?
4. What are some applications of the quadratic formula in real-world problems?
5. How do complex solutions arise in quadratic equations?

**Tip:** Always check if a cubic equation can be factored using identities like the sum or difference of cubes, which can simplify your work considerably.

Explore the step-by-step solution to solving the cubic equation (x-7)^3 + (3x+5)^3(2-4)^3 = 0. This problem involves algebraic concepts like the difference of cubes and solving quadratic equations using the quadratic formula. Suitable for Grades 10-12 students.

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