How to make non-linear regression equation with 4 variables? If we use only X and y, we can write y = mx + c equation for linear regression. 
But now I have another 2 variables. So now I have 4 variables. 
Let's think "x, y, z, w" as my 4 variables. How can I write the equation? If we use linear regression, we can use y = mx + c formula. but it will only represent x and y. but I want to incorporate all x, y, z, and w all together in one formula. 
What will be the equation?
 A: There's a few things to clear up here:


*

*With a single predictor ("independent variable") linear regression is sometimes referred to as simple linear regression. 
However, the formula "$y=mx+c$" doesn't correctly describe the relationship between $y$ and $x$; it may be that none of the y-values lay on that $mx+c$ line, so we need something else to describe how the values are related. 
We might write $y=mx+c+\epsilon$ (with $\epsilon$ generally a zero-mean error term) to describe the relationship between the $y$ and $x$ data, or we might put $E(Y|x)=mx+c$, where $Y$ here is the random variable that we're observing at $x$ and $E$ is its expectation. In both those cases, $m$ and $c$ are not our estimated slope and intercept, but the quantities being estimated. (Conventionally we'd then use $\hat{m}$ for the estimated slope)

*The linear in linear regression usually refers to linearity in the parameters, where "linear" takes the sense of its meaning in linear algebra rather than in (say) calculus or in reference to a linear polynomial. 

*If you want to extend simple linear regression to multiple predictors while still retaining linearity (in the linear algebra sense) then you will get multiple linear regression, which we might write as $y=\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_3+...+\beta_p x_p + \epsilon$ (we usually number the coefficients and predictors for a couple of reasons - one being so we don't run out of letters so fast) where $\epsilon$ is again the error term. It is common to write this in matrix form as $y=X\beta+\epsilon$. 
In terms of your response and three predictor variables you might write $y=c+m_xx+m_ww+m_zz+\epsilon$ (going back to your notation but extending it as needed).
With two predictors, the fitted relationship $\hat{y}=\hat{\beta}_0+\hat{\beta}_1x_1+\hat{\beta}_2x_2+e$ is a plane, and with more predictors you end up with hyperplanes of the relevant dimension. It's all "linear regression" in the sense that we usually use the term in statistics.

*However you can extend simple regression in other ways, such as constructing a relationship that has non-linear components (which would then actually fit your title), such as $y=\beta_0+\beta_1x+g_1(w)+g_2(z)+\epsilon$ (which we call an additive model), or some more general nonlinear function $y=g(x,w,z)+\epsilon$, which is usually called 'nonlinear regression'. 
If you see more than one of the variables as the response, you get multivariate regression. 
Numerous other extensions (though not remotely an exhaustive list) are mentioned at the link in point 3.
Many posts on our site discuss multiple regression. Try the tag multiple-regression
A: In general, you just add on more linear terms.
In this case, assuming you are predicting y, the equation will be
y = ax + bz + cw + d
