How do you solve these types of quadratics with cubes and squares? please explain and give a few more examples

I did read Factoring quadratics: the weird case, but I didn't find any example that matches the one I'm currently struggling with.

Thank you. Your help is greatly appreciated.

Solving cubic equations is usually more advanced in scope than Intermediate Algebra. You should look for possible binomial factors as described in the topic of the Rational Roots Theorem.

Actually, you should factor the polynomial first, into 2(x^3-5+1). You would then want to check for potential zeros from { -1/2, -1, 1/2, 1 }. THIS would be better.

Read about and study the Rational Roots Theorem to find better how this works. You need to use at least polynomial division, if not know synthetic division.