One week after, we still don't know if this is homework or not.
Well... one week after, it might be stale. So here are a couple of
answers to a question interesting beyond the OP's potential homework.

The problem data can be set up in Sage as follows:

```
# Problem data
# Arguments of the function to optimize and nullity constraints to satisfy
Args = var("w, x, y, z")
# Function to optimize (here a symbolic expression)
F = (4*x + 4*y + 9*z - 2*w)
# Constraints (here, symbolic expressions to nullify)
G = (2*x + 2*y + z + w)
H = (x^2 + y^2 + z - 4)
# Problem setup: tuple of constraints -- expressed in
# a way that does not depend on the particular problem
C = (G, H)
```

### General solution

The general solution states that if (the $n$-dimensional vector) $\mathbf{x_0}$
maximizes (the scalar) $\phi(\mathbf{x})$ under the ($m$-vector) equality constraint
$\boldsymbol{\psi}\mathbf{(x}) = \mathbf{0}$, then there exist an ($m$-dimensional) vector
$\mathbf{\lambda}$ (the Lagrange multipliers) such as all the components of the
differential of the (scalar) function (callad Lagrangian)
$\phi(\mathbf{x_0})+ \mathbf{\lambda} \cdot \mathbf{\boldsymbol{\psi}(x_0)}$ are all zero;
in other words, the maximization of a scalar function of $n$ variables
under $m$ constraints is replaced by the maximization
of a scalar function of $n + m$ variables.

This translates in Sage as:

```
# Solution
# (Vector) function of constraints
psi = vector(C).function(*Args)
# Vector of Lagrange multipliers
Lambda = vector([var("lambda_{}".format(u)) for u in range(len(C))])
# Function to optimize
phi = F.function(*Args)
# Lagrangian arguments
ArgsL = Args + tuple(Lambda)
# Lagrangian function
L = (phi + Lambda.dot_product(psi)).function(*ArgsL)
# Its differential (vector)
dL = L.diff()(*ArgsL)
# Solution(s) nullifying this differential :
Sol = solve(list(dL), ArgsL, solution_dict=True)
# Extremal value
Max = F.subs(Sol)
```

which gives:

```
sage: Sol
[{w: -628/121, x: 4/11, y: 4/11, z: 452/121, lambda_0: 2, lambda_1: -11}]
sage: Max
516/11
```

**Nota bene:** the Lagrange multipliers solutions *can* be maxima or minima,
but may also be saddle points; this should be checked.

### Specialized solution

The idea is to use the $m$ constraints to express $m$ variables as functions of
the other $n - m$ variables. (Here, we can express $w$ and $z$ as functions
of $x$ and $y$). The problem is then to find the (unconstrained) maximum
of a function of $n - m$ variables, which is done by solving the components
of its differential for those variables.

This is called parameterization.

```
# Eliminate two variables from the two constraints. Here, we choose w and z,
# since one constraint uses x^2 and y^2, opening the door to multiple solutions
S1 = solve([G, H], [w, z], solution_dict=True)
# Substitute this in F
F1 = F.subs(S1)
# Optimize F1, i. e. find the point(s) where both derivatives are zero
Sol1 = solve([F1.diff(u) for u in (x, y)], [x, y], solution_dict=True)
# Substitute in the [G, H] system, and solve for (numerical) w, z
Sol2 = solve([u.subs(Sol1[0]) for u in (G, H)], [w, z], solution_dict=True)
# Cleanup: One dictionary solution
HSSol = copy(Sol1[0])
HSSol.update(Sol2[0])
# Get the (numerical) extremum
HSMax = F.subs(HSSol)
```

which gives:

```
sage: HSSol
{x: 4/11, y: 4/11, w: -628/121, z: 452/121}
sage: HSMax
516/11
```

Again, the nature (maximum, minimum or saddle-point)
of the extrema thus obtained must be checked.

### Numerical solutions

One can note that Sage implements numerical methods of minimization
of a real function subject to (more general inequality) constraints,
accessible via the `minimize_constrained`

function. Other methods
implemented in SciPy (which Sage depends on) can also be used.

Homework ?

Use Lagrange multipliers?

Where are you stuck?

Still homework or not ?

Hint : the maximum you seek is $\displaystyle\frac{516}{11}$...

Another hint : in your specific case, Lagrange multipliers are not strictly necessary, and a high school-level method is sufficient...

Nevertheless, looking up "Lagrange multipliers" in Wikipedia and understanding it is

stronglyadvisable, event if the rigorous proofs underlying it require a bit more analysis than high school-level... (Anyway, high school-level analysis postulates some properties of the real numbers, whose proof requires way more than high school-level algebra and topology (the latter being null IIRC...)).