Verify Equivalence: 5x²+7x+7 = (2x+3)(3x+4)

Are the expressions on both sides equivalent?

$5x^2+7x+7\stackrel{?}{=}(2x+3)(3x+4)$

To determine if the expressions are equivalent, we need to expand the right-side expression, $(2x + 3)(3x + 4)$, and compare it with $5x^2 + 7x + 7$.

Let's expand the right-side expression:

• First, use the distributive property (or FOIL):
• $(2x + 3)(3x + 4) = 2x \cdot 3x + 2x \cdot 4 + 3 \cdot 3x + 3 \cdot 4$
• This simplifies to $6x^2 + 8x + 9x + 12$.
• Combine like terms: $6x^2 + (8x + 9x) + 12 = 6x^2 + 17x + 12$.

Now, compare the expanded expression $6x^2 + 17x + 12$ to the left side $5x^2 + 7x + 7$:

• The coefficient of $x^2$ is 6 on the right, but 5 on the left.
• The coefficient of $x$ is 17 on the right, but 7 on the left.
• The constant term is 12 on the right, but 7 on the left.

Since all corresponding coefficients differ between the two sides, the expressions are not equivalent.

Therefore, the correct answer is: No, because all the coefficients of the corresponding terms in the expressions on both sides of the equation are different.