Question

find a simplified expression to represent the area of the triangle. the area formula for a triangle is \\(\frac{1}{2}bh\\), where \\(b\\) is the base and \\(h\\) is the height.
(2x + 12) cm
(4x - 2) cm
the expression that represents the area of this triangle is \\(\boldsymbol{\quad} x^2 + \boldsymbol{\quad} x + \boldsymbol{\quad} \\, \text{cm}^2\\).

Explanation:

Step1: Identify base and height

Base \( b = (4x - 2) \) cm, Height \( h = (2x + 12) \) cm.

Step2: Apply area formula

Area \( A=\frac{1}{2}bh=\frac{1}{2}(4x - 2)(2x + 12) \).

Step3: Simplify the product

First, factor out 2 from \( 4x - 2 \): \( 4x - 2 = 2(2x - 1) \).
So \( A=\frac{1}{2}\times2(2x - 1)(2x + 12)=(2x - 1)(2x + 12) \).
Expand \( (2x - 1)(2x + 12) \):

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Wait, no, earlier step: \( \frac{1}{2}(4x - 2)(2x + 12)=\frac{1}{2}(8x^{2}+48x - 4x - 24)=\frac{1}{2}(8x^{2}+44x - 24)=4x^{2}+22x - 12 \). Wait, but the problem's box is for \( \square x^{2}+\square x+\square \), but our constant term is -12. Wait, maybe I made a mistake in base/height? Wait, the diagram: the base is \( (4x - 2) \), height is \( (2x + 12) \). Wait, let's re - calculate \( \frac{1}{2}(4x - 2)(2x + 12) \):

Multiply \( (4x - 2)(2x + 12) \):
\( 4x\times2x=8x^{2} \), \( 4x\times12 = 48x \), \( - 2\times2x=-4x \), \( -2\times12=-24 \). So \( 8x^{2}+48x - 4x - 24=8x^{2}+44x - 24 \). Then multiply by \( \frac{1}{2} \): \( \frac{1}{2}(8x^{2}+44x - 24)=4x^{2}+22x - 12 \). But the problem has a plus sign for the constant term. Wait, maybe the base and height are misidentified? Wait, maybe the base is \( (4x - 2) \) and height is \( (2x + 12) \), but let's check again. Wait, maybe I messed up the signs. Wait, \( (4x - 2)(2x + 12)=8x^{2}+48x-4x - 24 = 8x^{2}+44x - 24 \), half of that is \( 4x^{2}+22x - 12 \). But the problem's blanks are for \( \square x^{2}+\square x+\square \), so maybe there's a mistake in my calculation? Wait, no, let's do it again:

\( \frac{1}{2}(4x - 2)(2x + 12)=\frac{1}{2}[4x\times2x+4x\times12-2\times2x - 2\times12]=\frac{1}{2}[8x^{2}+48x - 4x - 24]=\frac{1}{2}[8x^{2}+44x - 24]=4x^{2}+22x - 12 \). So the coefficients are 4 (for \( x^{2} \)), 22 (for \( x \)), and - 12 (constant). But the problem's last box is a plus, maybe a typo, but according to the calculation, the simplified expression is \( 4x^{2}+22x - 12 \). Wait, maybe I misread the base and height. Wait, the height is \( (2x + 12) \), base is \( (4x - 2) \). Let me check the multiplication again:

\( (4x - 2)(2x + 12)=4x(2x)+4x(12)-2(2x)-2(12)=8x^{2}+48x - 4x - 24=8x^{2}+44x - 24 \). Then half of that is \( 4x^{2}+22x - 12 \). So the first blank is 4, second is 22, third is - 12. But maybe the problem has a different base/height? Wait, maybe the base is \( (4x + 2) \)? No, the diagram says \( (4x - 2) \). Alternatively, maybe the height is \( (2x - 12) \)? No, the diagram says \( (2x + 12) \). So according to the calculation, the coefficients are 4, 22, - 12.

Answer:

First box: 4, Second box: 22, Third box: - 12