1. **State the problem:** Factor the expression $$q^4 + 3q^2 - 10$$ into two binomials. 2. **Identify the structure:** Notice that $$q^4$$ is $$\left(q^2\right)^2$$, so treat $$q^2$$ as a single variable, say $$x$$. Then the expression becomes $$x^2 + 3x - 10$$. 3. **Factor the quadratic in terms of $$x$$:** We look for two numbers that multiply to $$-10$$ and add to $$3$$. These numbers are $$5$$ and $$-2$$. 4. **Write the factored form in terms of $$x$$:** $$x^2 + 3x - 10 = (x + 5)(x - 2)$$ 5. **Substitute back $$x = q^2$$:** $$ (q^2 + 5)(q^2 - 2) $$ 6. **Check if further factorization is possible:** - $$q^2 + 5$$ cannot be factored further over the real numbers. - $$q^2 - 2$$ is a difference of squares if written as $$q^2 - \sqrt{2}^2$$, so it factors as: $$ (q - \sqrt{2})(q + \sqrt{2}) $$ 7. **Final factorization:** $$ (q^2 + 5)(q - \sqrt{2})(q + \sqrt{2}) $$ **Answer:** $$q^4 + 3q^2 - 10 = (q^2 + 5)(q - \sqrt{2})(q + \sqrt{2})$$