Question:

<p><strong>Step 1:</strong> Find the derivative of <span style='font-style:italic'>f</span>(<span style='font-style:italic'>x</span>):</p><p>$$f'(x) = 7x^6 + 70x^4 + 48x^2 + 30$$</p><p><strong>Step 2:</strong> Analyze the sign of <span style='font-style:italic'>f</span>'(<span style='font-style:italic'>x</span>):</p><p>Since all coefficients are positive, we have $f'(x) > 0$ for all $x \in \mathbb{R}$.</p><p><strong>Step 3:</strong> Conclude:</p><p>Since <span style='font-style:italic'>f</span>'(<span style='font-style:italic'>x</span>) > 0 for all <span style='font-style:italic'>x</span>, the function <span style='font-style:italic'>f</span>(<span style='font-style:italic'>x</span>) is strictly increasing on $\mathbb{R}$. A strictly monotonic function crosses the x-axis at most once.</p><p>∴ The number of real solutions is <strong>1</strong>.</p>