If p,q,r,s are the roots of (x^2+x+4)^2 + 3x(x^2+x+4)+2x^2 = 0, then |p|+|q|+|r|+|s| is equal to :

To solve the equation \((x^2+x+4)^2 + 3x(x^2+x+4)+2x^2 = 0\) and find the sum of the absolute values of its roots, we can start by simplifying the expression. Let's denote \(y = x^2 + x + 4\). This substitution will help us rewrite the equation in a more manageable form.

Substituting and Simplifying

By substituting \(y\) into the equation, we have:

• \((y)^2 + 3x(y) + 2x^2 = 0\)

This can be expressed as:

• \(y^2 + 3xy + 2x^2 = 0\)

Quadratic in Terms of y

Now, we can treat this as a quadratic equation in \(y\):

• Using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 3x\), and \(c = 2x^2\), we find:

\[ y = \frac{-3x \pm \sqrt{(3x)^2 - 4(1)(2x^2)}}{2(1)} = \frac{-3x \pm \sqrt{9x^2 - 8x^2}}{2} = \frac{-3x \pm x}{2} \]

Finding y Values

This simplifies to two cases:

• Case 1: \(y = \frac{-2x}{2} = -x\)
• Case 2: \(y = \frac{-4x}{2} = -2x\)

Back to x Values

Now we substitute back \(y = x^2 + x + 4\) into both cases:

Case 1: \(y = -x\)

Setting \(x^2 + x + 4 = -x\) gives:

• \(x^2 + 2x + 4 = 0\)

The discriminant is \(2^2 - 4 \cdot 1 \cdot 4 = 4 - 16 = -12\), indicating two complex roots.

Case 2: \(y = -2x\)

Setting \(x^2 + x + 4 = -2x\) results in:

• \(x^2 + 3x + 4 = 0\)

The discriminant here is \(3^2 - 4 \cdot 1 \cdot 4 = 9 - 16 = -7\), which also indicates two complex roots.

Roots Summary

In total, we have four complex roots from both cases. To find the absolute values, we need to calculate the roots explicitly. The roots from the first case are:

• \(x = \frac{-2 \pm i\sqrt{12}}{2} = -1 \pm i\sqrt{3}\)

The absolute value of each root is:

• \(|-1 + i\sqrt{3}| = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2\)
• \(|-1 - i\sqrt{3}| = 2\)

For the second case, the roots are:

• \(x = \frac{-3 \pm i\sqrt{7}}{2}\)

The absolute value of each root is:

• \(|\frac{-3}{2} + i\frac{\sqrt{7}}{2}| = \sqrt{(\frac{-3}{2})^2 + (\frac{\sqrt{7}}{2})^2} = \sqrt{\frac{9}{4} + \frac{7}{4}} = \sqrt{4} = 2\)
• \(|\frac{-3}{2} - i\frac{\sqrt{7}}{2}| = 2\)

Calculating the Final Sum

Now, we can sum the absolute values of all four roots:

• From the first case: \(2 + 2 = 4\)
• From the second case: \(2 + 2 = 4\)

Thus, the total sum of the absolute values is:

|p| + |q| + |r| + |s| = 4 + 4 = 8

Therefore, the answer is 8.