Find the greatest value of a non-negative real number `lambda` for which both the equations `2x^2+(lambda-1)x+8=0a n dx^2-8x+lambda+4=0` have real roots.

Explanation

To find the greatest value of a non-negative real number \( \lambda \) for which both equations \( 2x^2 + (\lambda - 1)x + 8 = 0 \) and \( x^2 - 8x + (\lambda + 4) = 0 \) have real roots, we need to ensure that the discriminants of both quadratic equations are non-negative.

### Step 1: Analyze the first quadratic equation

The first equation is:
\[ 2x^2 + (\lambda - 1)x + 8 = 0 \]
The discriminant \( D_1 \) for this equation is given by:
\[ D_1 = b^2 - 4ac \]
where \( a = 2 \), \( b = \lambda - 1 \), and \( c = 8 \). Thus,
\[ D_1 = (\lambda - 1)^2 - 4 \cdot 2 \cdot 8 \]
Calculating this, we have:
\[ D_1 = (\lambda - 1)^2 - 64 \]
For the roots to be real, we require:
\[ D_1 \geq 0 \]
This leads to:
\[ (\lambda - 1)^2 \geq 64 \]
Taking the square root of both sides, we get:
\[ |\lambda - 1| \geq 8 \]
This results in two inequalities:
1. \( \lambda - 1 \geq 8 \) which simplifies to \( \lambda \geq 9 \)
2. \( \lambda - 1 \leq -8 \) which simplifies to \( \lambda \leq -7 \)

### Step 2: Analyze the second quadratic equation

The second equation is:
\[ x^2 - 8x + (\lambda + 4) = 0 \]
The discriminant \( D_2 \) for this equation is given by:
\[ D_2 = b^2 - 4ac \]
where \( a = 1 \), \( b = -8 \), and \( c = \lambda + 4 \). Thus,
\[ D_2 = (-8)^2 - 4 \cdot 1 \cdot (\lambda + 4) \]
Calculating this, we have:
\[ D_2 = 64 - 4(\lambda + 4) = 64 - 4\lambda - 16 = 48 - 4\lambda \]
For the roots to be real, we require:
\[ D_2 \geq 0 \]
This leads to:
\[ 48 - 4\lambda \geq 0 \]
Rearranging gives:
\[ 48 \geq 4\lambda \quad \Rightarrow \quad \lambda \leq 12 \]

### Step 3: Combine the results

From the first quadratic, we have:
- \( \lambda \geq 9 \) or \( \lambda \leq -7 \)

From the second quadratic, we have:
- \( \lambda \leq 12 \)

Since we are looking for non-negative values of \( \lambda \), we can ignore \( \lambda \leq -7 \).

Thus, we combine:
\[ 9 \leq \lambda \leq 12 \]

### Step 4: Find the greatest value of \( \lambda \)

The greatest non-negative value of \( \lambda \) that satisfies both conditions is:
\[ \lambda = 12 \]

### Final Answer:
The greatest value of a non-negative real number \( \lambda \) for which both equations have real roots is:
\[ \boxed{12} \]