Simplify x2 minus 3 times the square of (x/2 plus the square root of 5).

Step-by-step solution

1. 1

   Square the bracket first — the exponent binds tighter than the 3. Using $(a+b)^2=a^2+2ab+b^2$ with $a=\frac{x}{2}$ and $b=\sqrt5$:

   $a^2=\frac{x^2}{4},\qquad 2ab=2\cdot\frac{x}{2}\cdot\sqrt5=x\sqrt5,\qquad b^2=(\sqrt5)^2=5$

   so $\left(\frac{x}{2}+\sqrt5\right)^2=\frac{x^2}{4}+x\sqrt5+5$. Note $b^2=5$ exactly — squaring removes the radical.

2. 2

   Multiply the whole square by 3.

   $3\left(\frac{x^2}{4}+x\sqrt5+5\right)=\frac{3x^2}{4}+3x\sqrt5+15$

   Every one of the three terms is tripled, the surd term included.

3. 3

   Subtract, distributing the minus sign across all three terms.

   $x^2-\left(\frac{3x^2}{4}+3x\sqrt5+15\right)=x^2-\frac{3x^2}{4}-3x\sqrt5-15$

   This is where a dropped sign usually appears; the middle and last terms both become negative.

4. 4

   Combine the $x^2$ terms.

   $x^2-\frac{3x^2}{4}=\frac{4x^2-3x^2}{4}=\frac{x^2}{4}$

   so the simplified expression is

   $\frac{x^2}{4}-3x\sqrt5-15$

5. 5

   Check at $x=2$. The original is $4-3\left(1+\sqrt5\right)^2=4-3\left(6+2\sqrt5\right)=4-18-6\sqrt5=-14-6\sqrt5\approx -27.416$. The answer gives $1-6\sqrt5-15=-14-6\sqrt5\approx -27.416$ ✓ (using $3\sqrt5\approx 6.7082$).