\( 1 \leftarrow \quad \begin{array}{l}\text { Factor, Check by multiplying, } \\ 10 x^{7}=9 x^{5}+18 x^{4} \\ 10 x^{7}-9 x^{5}+18 x^{4} \equiv \square\end{array} \)

To factor the expression \( 10x^{7} - 9x^{5} + 18x^{4} \), we can start by looking for common factors in the terms. The greatest common factor is \( x^4 \), so we factor that out: \[ x^4(10x^{3} - 9x + 18) \] Now, we can work on simplifying \( 10x^{3} - 9x + 18 \), but for now, let's just note this expression. To check the factorization, we can multiply it back out to ensure we return to the original polynomial. Multiplying gives us \( 10x^{7} - 9x^{5} + 18x^{4} \), confirming that our factorization is correct as \( x^4(10x^{3} - 9x + 18) \). Look closely at the expression after factoring. This is a polynomial, and polynomial identities like the one we've just noticed can often lead to further factorizations or yield roots that may not be immediately obvious. Bringing in graphing technology can also assist in identifying the nature of the polynomial's roots and understanding its behavior better! If you're interested in exploring more about cubic polynomials and factoring techniques, various resources discuss the Rational Root Theorem, synthetic division, and polynomial long division. Combining these tools can equip you for more advanced algebraic problems!