Simplify the expression: \( |5x^2 - 10x + 3| \)

Step 1 ：Step 1: Factor the polynomial inside the absolute value. Here we have \( |5x^2 - 10x + 3| = |5(x^2-2x)+3| \)

Step 2 ：Step 2: Complete the square inside the absolute value. \( |5(x^2-2x)+3| = |5[(x-1)^2 - 1] + 3| = |5(x-1)^2 - 5 + 3| = |5(x-1)^2 - 2| \)

Step 3 ：Step 3: An absolute value is always positive, so we have \( |5(x-1)^2 - 2| \geq 0 \). This means the expression is equal to itself when \( 5(x-1)^2 - 2 \geq 0 \) and equal to its opposite when \( 5(x-1)^2 - 2 < 0 \).

Step 4 ：Step 4: Solve for x in each case. If \( 5(x-1)^2 - 2 \geq 0 \), then \( x \leq 1 - \sqrt{2/5} \) or \( x \geq 1 + \sqrt{2/5} \). If \( 5(x-1)^2 - 2 < 0 \), then \( 1 - \sqrt{2/5} < x < 1 + \sqrt{2/5} \). So the simplified expression is \( 5(x-1)^2 - 2 \) when \( x \leq 1 - \sqrt{2/5} \) or \( x \geq 1 + \sqrt{2/5} \), and \( -5(x-1)^2 + 2 \) when \( 1 - \sqrt{2/5} < x < 1 + \sqrt{2/5} \).