https://file.aiquickdraw.com/mathbot/file/17299881311140rj9nshn.jpg

The equation in the image is:

\[
x^4 - 8x^2 + 2x^3 + 38 = 2 + 2x^3 + 5x^2
\]

To solve this, let’s start by simplifying and rearranging terms.

### Step 1: Move all terms to one side of the equation

Subtract \(2 + 2x^3 + 5x^2\) from both sides:

\[
x^4 - 8x^2 + 2x^3 + 38 - 2 - 2x^3 - 5x^2 = 0
\]

This simplifies to:

\[
x^4 - 13x^2 + 36 = 0
\]

### Step 2: Solve the simplified equation

Now we have a quadratic equation in terms of \(x^2\):
\[
x^4 - 13x^2 + 36 = 0
\]

Let \(u = x^2\), then \(u^2 = x^4\). Substitute \(u\) into the equation:

\[
u^2 - 13u + 36 = 0
\]

### Step 3: Solve for \(u\) using the quadratic formula

The quadratic formula is given by:

\[
u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this case, \(a = 1\), \(b = -13\), and \(c = 36\). Substitute these values:

\[
u = \frac{13 \pm \sqrt{(-13)^2 - 4 \cdot 1 \cdot 36}}{2 \cdot 1}
\]
\[
u = \frac{13 \pm \sqrt{169 - 144}}{2}
\]
\[
u = \frac{13 \pm \sqrt{25}}{2}
\]
\[
u = \frac{13 \pm 5}{2}
\]

This gives two solutions for \(u\):

1. \( u = \frac{13 + 5}{2} = 9 \)
2. \( u = \frac{13 - 5}{2} = 4 \)

### Step 4: Substitute back for \(x\)

Since \(u = x^2\), we have:

1. \( x^2 = 9 \Rightarrow x = \pm 3 \)
2. \( x^2 = 4 \Rightarrow x = \pm 2 \)

### Final Answer

The solutions for \(x\) are:
\[
x = \pm 3, \pm 2
\]

---

Would you like more details or have any questions about the steps?

### Related Questions:
1. How would you verify each solution for \(x\) in the original equation?
2. What other methods could you use to solve a polynomial like this one?
3. How does substituting \(u = x^2\) simplify solving higher-degree polynomials?
4. What would the graph of \(x^4 - 13x^2 + 36 = 0\) look like?
5. How can we determine if an equation is factorable or requires the quadratic formula?

### Tip:
When solving polynomial equations, look for substitutions or factorable patterns to reduce the degree and simplify solving.

Solve the equation x^4 - 8x^2 + 2x^3 + 38 = 2 + 2x^3 + 5x^2

Learn how to solve the polynomial equation x^4 - 8x^2 + 2x^3 + 38 = 2 + 2x^3 + 5x^2. This solution involves simplifying, rearranging, and using substitution to reduce the equation to a quadratic form, then solving with the quadratic formula. Suitable for students in Grades 10-12.

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