1. The problem is to express the polynomial $9x^{10} - 15x^4 + 9$ in quadratic form if possible. 2. A quadratic form is generally an expression of the form $ax^2 + bx + c$ where the highest power of $x$ is 2. 3. Here, the polynomial has terms with powers 10, 4, and 0 (constant term), which are not powers of 2 or less. 4. To check if it can be rewritten as a quadratic in some variable substitution, let us try substituting $y = x^n$ for some $n$ such that the powers become multiples of 2. 5. The powers are 10 and 4. The greatest common divisor (GCD) of 10 and 4 is 2. 6. Let $y = x^2$. Then $x^{10} = (x^2)^5 = y^5$ and $x^4 = (x^2)^2 = y^2$. 7. Substituting, the polynomial becomes $9y^5 - 15y^2 + 9$. 8. This is a polynomial in $y$ of degree 5, which is not quadratic. 9. Since the polynomial in $y$ is degree 5, it cannot be expressed as a quadratic polynomial in $y$. 10. Therefore, the original polynomial cannot be put into quadratic form by any substitution. Final answer: The polynomial $9x^{10} - 15x^4 + 9$ cannot be expressed in quadratic form because its terms correspond to powers that do not reduce to degree 2 or less under any substitution.