If the equation `(m^(2) -12 )x^(4) -8x ^(2)-4=0` has no real roots, then the largest value of m is `psqrtq` whre p, q are coprime natural numbers, then `p +q=`

Answer

To solve the problem, we need to analyze the given equation:

\[ (m^2 - 12)x^4 - 8x^2 - 4 = 0 \]

### Step 1: Substitute \( x^2 \) with \( a \)

Let \( a = x^2 \). Then, the equation becomes:

\[ (m^2 - 12)a^2 - 8a - 4 = 0 \]

### Step 2: Identify the coefficients

In this quadratic equation in terms of \( a \), we have:
- \( A = m^2 - 12 \)
- \( B = -8 \)
- \( C = -4 \)

### Step 3: Condition for no real roots

For the quadratic equation \( Aa^2 + Ba + C = 0 \) to have no real roots, the discriminant \( D \) must be less than 0:

\[ D = B^2 - 4AC < 0 \]

Substituting the values of \( A \), \( B \), and \( C \):

\[ (-8)^2 - 4(m^2 - 12)(-4) < 0 \]

### Step 4: Simplify the discriminant

Calculating the discriminant:

\[ 64 + 16(m^2 - 12) < 0 \]

This simplifies to:

\[ 64 + 16m^2 - 192 < 0 \]

\[ 16m^2 - 128 < 0 \]

### Step 5: Further simplify the inequality

Dividing the entire inequality by 16:

\[ m^2 - 8 < 0 \]

### Step 6: Solve the inequality

This can be rewritten as:

\[ m^2 < 8 \]

Taking the square root of both sides gives:

\[ -\sqrt{8} < m < \sqrt{8} \]

### Step 7: Determine the largest value of \( m \)

The largest value of \( m \) that satisfies this inequality is:

\[ m = \sqrt{8} = 2\sqrt{2} \]

### Step 8: Identify \( p \) and \( q \)

Here, \( p = 2 \) and \( q = 2 \). Since \( p \) and \( q \) are coprime natural numbers, we can find \( p + q \):

\[ p + q = 2 + 2 = 4 \]

### Final Answer

Thus, the answer is:

\[ \boxed{4} \]