Expand `(y+x)^2`

Answer

To expand \((y+x)^2\), we can use the binomial theorem or simply apply the algebraic identity for the square of a binomial. Here’s the step-by-step solution:

### Step 1: Identify the expression
We start with the expression \((y + x)^2\).

### Step 2: Apply the binomial expansion formula
The binomial expansion formula for \((a + b)^n\) is given by:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
For our case, \(a = y\), \(b = x\), and \(n = 2\).

### Step 3: Substitute the values into the formula
Using the formula, we can expand \((y + x)^2\):
\[ (y + x)^2 = \binom{2}{0} y^2 x^0 + \binom{2}{1} y^1 x^1 + \binom{2}{2} y^0 x^2 \]

### Step 4: Calculate the binomial coefficients
Now we calculate the binomial coefficients:
- \(\binom{2}{0} = 1\)
- \(\binom{2}{1} = 2\)
- \(\binom{2}{2} = 1\)

### Step 5: Substitute the coefficients back into the expression
Now substituting the coefficients back into the expression:
\[ (y + x)^2 = 1 \cdot y^2 \cdot 1 + 2 \cdot y \cdot x + 1 \cdot 1 \cdot x^2 \]

### Step 6: Simplify the expression
This simplifies to:
\[ (y + x)^2 = y^2 + 2xy + x^2 \]

### Final Answer
Thus, the expansion of \((y + x)^2\) is:
\[ y^2 + 2xy + x^2 \]

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