Express $$ x^{2}+x $$ in the form $$ a x(x+2)+b(x+3)+c $$

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UpStudy AI · Accepted Answer

To express $$ x^{2} + x $$ in the form $$ a x(x+2) + b(x+3) + c $$, let's determine the constants $$ a $$, $$ b $$, and $$ c $$.

1. **Start with the equation:**
   $$
   x^{2} + x = a x(x + 2) + b(x + 3) + c
   $$

2. **Expand the right-hand side:**
   $$
   a x(x + 2) = a x^{2} + 2a x
   $$
   $$
   b(x + 3) = b x + 3b
   $$
   So,
   $$
   a x^{2} + (2a + b) x + 3b + c
   $$

3. **Set up the system of equations by equating coefficients:**
   - For $$ x^{2} $$:
     $$
     a = 1
     $$
   - For $$ x $$:
     $$
     2a + b = 1 \quad \Rightarrow \quad 2(1) + b = 1 \quad \Rightarrow \quad b = -1
     $$
   - For the constant term:
     $$
     3b + c = 0 \quad \Rightarrow \quad 3(-1) + c = 0 \quad \Rightarrow \quad c = 3
     $$

4. **Substitute the values of $$ a $$, $$ b $$, and $$ c $$ back into the expression:**
   $$
   x^{2} + x = 1 \cdot x(x + 2) - 1(x + 3) + 3
   $$
   Simplifying:
   $$
   x^{2} + x = x(x + 2) - (x + 3) + 3
   $$

**Final Answer:**
$$
x^{2} + x = x(x + 2) - (x + 3) + 3
$$
$$ x^{2} + x = x(x + 2) - (x + 3) + 3 $$

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