The differential equation $$ \frac{d y}{d x}=10+8 x+20 y+16 x y $$ has an implicit general solution of the form $$ F(x, y)=K $$, where $$ K $$ is an arbitrary constant. In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form $$ F(x, y)=G(x)+H(y)=K . $$ Find such a solution and then give the related functions requested. $$ F(x, y)=G(x)+H(y)= $$

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We start with the differential equation
$$
\frac{d y}{d x} = 10 + 8x + 20 y + 16xy.
$$
Notice that the right‐hand side factors as follows:
$$
10 + 8x + 20y + 16xy = (10 + 8x) + (20y + 16xy) = (10 + 8x)(1 + 2y).
$$
Thus, the differential equation becomes
$$
\frac{dy}{dx} = (10+8x)(1+2y).
$$

Since the equation is separable, we separate the variables by writing
$$
\frac{dy}{1+2y} = (10+8x) \, dx.
$$

Next, we integrate both sides.

### Integrating the Left-Hand Side

Let
$$
u = 1+2y \quad \Longrightarrow \quad du = 2\,dy \quad \Longrightarrow \quad dy = \frac{du}{2}.
$$
Then,
$$
\int \frac{dy}{1+2y} = \int \frac{\frac{du}{2}}{u} = \frac{1}{2} \int \frac{du}{u} = \frac{1}{2}\ln|u| = \frac{1}{2}\ln|1+2y|.
$$

### Integrating the Right-Hand Side

We have
$$
\int (10+8x)\, dx = 10x + 4x^2,
$$
where the constant of integration is omitted at this stage.

### Combining the Integrals

Equate the results:
$$
\frac{1}{2}\ln|1+2y| = 10x + 4x^2 + C.
$$
Multiplying both sides by 2 gives
$$
\ln|1+2y| = 20x + 8x^2 + C_1,
$$
where $$C_1 = 2C$$ is an arbitrary constant. We can absorb $$C_1$$ into an arbitrary constant $$K$$ when writing the solution implicitly. That is, we write
$$
\ln|1+2y| - 20x - 8x^2 = K.
$$

### Expressing the Answer in the Form $$ F(x,y)=G(x)+H(y)=K $$

In our implicit general solution, we identify
$$
G(x) = -20x - 8x^2 \quad \text{and} \quad H(y) = \ln|1+2y|.
$$
Thus, the solution can be expressed as
$$
F(x,y) = G(x) + H(y) = \ln|1+2y| - 20x - 8x^2 = K.
$$
$$ F(x, y) = \ln|1 + 2y| - 20x - 8x^2 = K $$

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