The formula for the Sum of Squares

To solve this problem, we will use the formula for the square of a sum, which is $(a+b)^2 = a^2 + 2ab + b^2$.

Here, we identify $a = x$ and $b = y$. Thus, our expression $(x+y)^2$ can be expanded as follows:

$(x+y)^2 = x^2 + 2xy + y^2$

Now we will match this expanded form with the given choices.

• Choice 1: $x^2 + y^2$ - This is missing the middle term $2xy$.
• Choice 2: $y^2 + x^2 + 2xy$ - This matches the expanded expression.
• Choice 3: $x^2 + xy + y^2$ - The term $xy$ doesn't match the required $2xy$.
• Choice 4: $x^2 - 2xy + y^2$ - The middle term has an incorrect sign.

Therefore, the expression that is equivalent to $(x+y)^2$ is choice 2: $y^2 + x^2 + 2xy$.

We use the abbreviated multiplication formula:

$x^2+2\times x\times3+3^2=$

$x^2+6x+9$

In this task, we are asked to simplify the formula using the abbreviated multiplication formulas.

Let's take a look at the formulas:

$(x-y)^2=x^2-2xy+y^2$

 $(x+y)^2=x^2+2xy+y^2$

$(x+y)\times(x-y)=x^2-y^2$

Taking into account that in the given exercise there is only addition operation, the appropriate formula is the second one:

Now let us consider, what number when multiplied by itself will equal 4 and what number when multiplied by itself will equal 25?

The answers are respectively 2 and 5:

We insert these into the formula:

$(2x+5)^2=$

$(2x+5)(2x+5)=$

$2x\times2x+2x\times5+2x\times5+5\times5=$

$4x^2+20x+25$

That means our solution is correct.

To solve the expression $(x^2 + 4)^2$, we will follow these steps:

• Step 1: Identify the expression as a binomial $(a + b)$, where $a = x^2$ and $b = 4$.
• Step 2: Apply the binomial square formula: $(a + b)^2 = a^2 + 2ab + b^2$.
• Step 3: Substitute $a$ and $b$ into the formula.
• Step 4: Calculate each term in the formula.
• Step 5: Simplify to arrive at the final expanded form.

Let's execute these steps:
Step 1: Identify $a = x^2$ and $b = 4$.
Step 2: Apply the formula $(a + b)^2 = a^2 + 2ab + b^2$.
Step 3: Substitute to get: $(x^2)^2 + 2(x^2)(4) + 4^2$.
Step 4: Calculate each term:
- $(x^2)^2 = x^4$,
- $2(x^2)(4) = 8x^2$,
- $4^2 = 16$.
Step 5: Combine the terms to get the expanded expression: $x^4 + 8x^2 + 16$.

Therefore, the solution to the expression is $x^4 + 8x^2 + 16$.

To solve this problem, follow these steps:

• Step 1: Identify the terms: Here, $a = x$ and $b = x^2$.
• Step 2: Apply the square of a binomial formula: $(a + b)^2 = a^2 + 2ab + b^2$.
• Step 3: Substitute the terms into the formula and simplify the resulting expression.

Now, let's work through each step:

Step 1: We have $a = x$ and $b = x^2$.
Step 2: The formula gives us: $(a + b)^2 = a^2 + 2ab + b^2$

Step 3: Substituting $a = x$ and $b = x^2$ into the formula:

This simplifies to:

Therefore, the expanded form of the expression $(x + x^2)^2$ is $\mathbf{x^2 + 2x^3 + x^4}$.

This matches with choice 3 from the options provided.