To solve the equation
\[ -\sqrt{-7x + 4} = \sqrt{1 - x} + 3, \]
we first isolate the square root on one side by adding \(\sqrt{-7x + 4}\) and \(\sqrt{1 - x}\) to both sides:
\[ 0 = \sqrt{1 - x} + 3 + \sqrt{-7x + 4}. \]
Now, let's rearrange it to:
\[ \sqrt{-7x + 4} = -\sqrt{1 - x} - 3. \]
Since the square root on the left side is non-negative, the right side must also be non-negative. Thus:
\[ -\sqrt{1 - x} - 3 \geq 0, \]
which implies
\[ -\sqrt{1 - x} \geq 3 \quad \Rightarrow \quad \sqrt{1 - x} \leq -3, \]
which is impossible, because square roots are always non-negative. This indicates that either:
- There are no solutions, or
- We made a mistake in how we isolated the terms.
Let's square both sides at some earlier point and see if we can solve it that way.
Starting from the original equation:
\[ -\sqrt{-7x + 4} = \sqrt{1 - x} + 3, \]
we can square both sides. Square both sides gives:
\[ (-\sqrt{-7x + 4})^2 = (\sqrt{1 - x} + 3)^2. \]
This simplifies to:
\[ -7x + 4 = (1 - x) + 6\sqrt{1 - x} + 9. \]
Updating gives:
\[ -7x + 4 = 10 - x + 6\sqrt{1 - x}. \]
Rearranging further,
\[ -7x + x + 4 - 10 = 6\sqrt{1 - x} \implies -6x - 6 = 6\sqrt{1 - x}. \]
Dividing by 6 yields:
\[ -x - 1 = \sqrt{1 - x}. \]
Again, we square both sides:
\[ (-x - 1)^2 = 1 - x. \]
Expanding gives:
\[ x^2 + 2x + 1 = 1 - x. \]
Bringing everything to one side gives:
\[ x^2 + 3x = 0. \]
Factoring out \(x\):
\[ x(x + 3) = 0. \]
Thus, we have two potential solutions:
\[ x = 0 \quad \text{or} \quad x = -3. \]
Now, we need to check whether these solutions satisfy the original equation:
For \(x = 0\)
\[ -\sqrt{-7(0) + 4} = \sqrt{1 - 0} + 3 \implies -\sqrt{4} = 1 + 3 \implies -2 = 4 \quad \text{(false)}. \]
For \(x = -3\)
\[ -\sqrt{-7(-3) + 4} = \sqrt{1 - (-3)} + 3 \implies -\sqrt{21 + 4} = \sqrt{4} + 3 \implies -\sqrt{25} = 2 + 3 \implies -5 = 5 \quad \text{(false)}. \]
Since neither solution satisfies the original equation, the equation \(-\sqrt{-7x + 4} = \sqrt{1 - x} + 3\) has no solutions in real numbers.
Thus, the answer to the original equation is:
\[ \boxed{\text{No solutions}}. \]