Algebraic Proof of Inequality
<p>Consider the quadratic expression \(1 - 3p^2 + 4p^3\).</p>
<p>We can rewrite it as \(4p^3 - 3p^2 + 1\).</p>
<p>Factoring by grouping, we get \(p^2(4p - 3) + 1\).</p>
<p>Notice that \(p^2\) is always non-negative, because any number squared is non-negative.</p>
<p>The term \(4p - 3\) can be negative, zero, or positive depending on the value of \(p\). However, since it is being multiplied by \(p^2\), which is non-negative, the whole term \(p^2(4p - 3)\) will never be less than \(-3\).</p>
<p>Adding \(1\) to \(p^2(4p - 3)\), we ensure that the expression will be greater than zero for all values of \(p\) because the minimum value of \(p^2(4p - 3)\) is \(-3\), and adding \(1\) to it gives us \(-2\), which is still greater than zero.</p>
<p>Therefore, we can conclude that \(1 - 3p^2 + 4p^3 > 0\) for all values of \(p\).</p>
