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Find the factor of this perfect square trinomials

To find the factors of the given perfect square trinomial, we first need to identify the trinomial from the image. The image shows:

\[ 20x^4 - 35x^2 + 18 \]

To factor this trinomial, let's proceed step by step.

1. **Check if the trinomial can be factored as a perfect square trinomial:**

A perfect square trinomial is of the form \( (ax^2 + bx + c)^2 \). However, our given trinomial does not fit this form directly. So, we should check if it can be factored by other means.

2. **Factorization:**

We will use substitution to simplify the trinomial. Let \( y = x^2 \). Then, the given trinomial becomes:
\[ 20y^2 - 35y + 18 \]

Next, we need to factor \( 20y^2 - 35y + 18 \). We look for two numbers that multiply to \( 20 \times 18 = 360 \) and add up to \( -35 \).

The numbers that satisfy these conditions are \( -20 \) and \( -15 \) because:
\[ -20 \times -15 = 300 \]
\[ -20 + -15 = -35 \]

Thus, we rewrite the trinomial as:
\[ 20y^2 - 20y - 15y + 18 \]

Next, we group the terms:
\[ 20y^2 - 20y - 15y + 18 = (20y^2 - 20y) + (-15y + 18) \]

Now, we factor out the greatest common factor (GCF) from each group:
\[ = 20y(y - 1) - 3(5y - 6) \]

Notice that the expression inside the parentheses is not the same, so we need to find another approach.

Since direct factoring is challenging here, we should check if the trinomial can be solved using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Where \( a = 20 \), \( b = -35 \), and \( c = 18 \). Substituting these values in, we get:
\[ y = \frac{35 \pm \sqrt{(-35)^2 - 4(20)(18)}}{2(20)} \]
\[ y = \frac{35 \pm \sqrt{1225 - 1440}}{40} \]
\[ y = \frac{35 \pm \sqrt{-215}}{40} \]

The discriminant is negative (\( -215 \)), indicating that the trinomial does not factor into real numbers.

Thus, \( 20x^4 - 35x^2 + 18 \) does not factor into real numbers and is not a perfect square trinomial. 

Would you like further details on solving such problems or have any questions?

### Relative Questions:
1. What is a perfect square trinomial?
2. How do you factorize a quadratic equation?
3. What does it mean when a discriminant is negative?
4. Can a trinomial be factored if the discriminant is negative?
5. How is the quadratic formula used in factorization?
6. Are there methods other than the quadratic formula to factorize trinomials?
7. What is substitution in algebra?
8. How can we verify if a trinomial is a perfect square?

### Tip:
When factoring trinomials, always check if the quadratic equation can be simplified using substitution to make the factorization process easier.

Learn how to factor perfect square trinomials with a detailed step-by-step solution. This problem explores factorization methods using substitution and the quadratic formula. Suitable for high school students studying algebra.

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